{"id":158,"date":"2017-05-11T17:08:02","date_gmt":"2017-05-11T17:08:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/chapter\/chapter-2-sampling-distributions-and-confidence-intervals\/"},"modified":"2017-05-11T18:19:19","modified_gmt":"2017-05-11T18:19:19","slug":"chapter-2-sampling-distributions-and-confidence-intervals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/chapter\/chapter-2-sampling-distributions-and-confidence-intervals\/","title":{"raw":"Chapter 2: Sampling Distributions and Confidence Intervals","rendered":"Chapter 2: Sampling Distributions and Confidence Intervals"},"content":{"raw":"<div class=\"Basic-Text-Frame\">\r\n<h2 class=\"Chapter-Number\">Sampling Distribution of the Sample Mean<\/h2>\r\nInferential testing uses the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) to estimate the population mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>). Typically, we use the data from a single sample, but there are many possible samples of the same size that could be drawn from that population. As we saw in the previous chapter, the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) is a random variable with its own distribution.\r\n<ul>\r\n \t<li class=\"List-Paragraph\">The distribution of the sample mean will have a mean equal to \u00b5.<\/li>\r\n \t<li class=\"List-Paragraph\">It will have a standard deviation (standard error) equal to <img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170640\/3341.png\" alt=\"3341.png\" width=\"35\" height=\"32\" \/><\/li>\r\n<\/ul>\r\nBecause our inferences about the population mean rely on the sample mean, we focus on the distribution of the sample mean. Is it normal? What if our population is not normally distributed or we don\u2019t know anything about the distribution of our population?\r\n\r\n<strong>The Central Limit Theorem<\/strong> states that the sampling distribution of the sample means will approach a normal distribution as the sample size increases.\r\n<ul>\r\n \t<li class=\"List-Paragraph\">So if we do not have a normal distribution, or know nothing about our distribution, the CLT tells us that the distribution of the sample means (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) will become normal distributed as <em>n<\/em> (sample size) increases.<\/li>\r\n \t<li class=\"List-Paragraph\">How large does <em>n<\/em> have to be?<\/li>\r\n \t<li class=\"List-Paragraph\">A general rule of thumb tells us that <em>n<\/em> \u2265 30.<\/li>\r\n<\/ul>\r\nThe Central Limit Theorem tells us that regardless of the shape of our population, the sampling distribution of the sample mean will be normal as the sample size increases.\r\n<h3>Sampling Distribution of the Sample Proportion<\/h3>\r\nThe population proportion (<em>p<\/em>) is a parameter that is as commonly estimated as the mean. It is just as important to understand the distribution of the sample proportion, as the mean. With proportions, the element either has the characteristic you are interested in or the element does not have the characteristic. The sample proportion (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>p\u0302<\/em><\/span>) is calculated by\r\n<p class=\"Centered\"><span class=\"Inline-Equation-Large\"><img class=\"frame-43 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170641\/3316.png\" alt=\"3316.png\" \/><\/span><\/p>\r\nwhere <em>x<\/em> is the number of elements in your population with the characteristic and <em>n<\/em> is the sample size.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\n<p class=\"Example\">You are studying the number of cavity trees in the Monongahela National Forest for wildlife habitat. You have a sample size of n = 950 trees and, of those trees, x = 238 trees with cavities. The sample proportion is:<\/p>\r\n<p class=\"ExampleCenter\"><img class=\"frame-44\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170642\/3305.png\" alt=\"3305.png\" \/><\/p>\r\n\r\n<\/div>\r\nThe distribution of the sample proportion has a mean of <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170643\/3295.png\" alt=\"3295.png\" width=\"55\" height=\"21\" \/><\/span>\r\n\r\nand has a standard deviation of <span class=\"Inline-Equation-Large\"><img class=\"frame-45\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170644\/3287.png\" alt=\"3287.png\" width=\"121\" height=\"50\" \/><\/span> .\r\n\r\nThe sample proportion is normally distributed if <em>n<\/em> is very large and <span class=\"Inline-Equation\"><img class=\"frame-5\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170645\/3282.png\" alt=\"3282.png\" width=\"10\" height=\"19\" \/><\/span> isn\u2019t close to 0 or 1. We can also use the following relationship to assess normality when the parameter being estimated is <em>p<\/em>, the population proportion:\r\n<p class=\"Centered\"><img class=\"frame-46 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170646\/3273.png\" alt=\"3273.png\" \/><\/p>\r\n\r\n<h2>Confidence Intervals<\/h2>\r\nIn the preceding chapter we learned that populations are characterized by descriptive measures called parameters. Inferences about parameters are based on sample statistics. We now want to estimate population parameters and assess the reliability of our estimates based on our knowledge of the sampling distributions of these statistics.\r\n<h3>Point Estimates<\/h3>\r\nWe start with a point estimate. This is a single value computed from the sample data that is used to estimate the population parameter of interest.\r\n<ul>\r\n \t<li class=\"List-Paragraph\">The sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) is a point estimate of the population mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>).<\/li>\r\n \t<li class=\"List-Paragraph\">The sample proportion (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>p\u0302<\/em><\/span>) is the point estimate of the population proportion (<em>p<\/em>).<\/li>\r\n<\/ul>\r\nWe use point estimates to construct confidence intervals for unknown parameters.\r\n<ul>\r\n \t<li class=\"List-Paragraph\">A confidence interval is an interval of values instead of a single point estimate.<\/li>\r\n \t<li class=\"List-Paragraph\">The level of confidence corresponds to the expected proportion of intervals that will contain the parameter if many confidence intervals are constructed of the same sample size from the same population.<\/li>\r\n \t<li class=\"List-Paragraph\">Our uncertainty is about whether our particular confidence interval is one of those that truly contains the true value of the parameter.<\/li>\r\n<\/ul>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\n<p class=\"Example\">We are 95% confident that our interval contains the population mean bear weight.<\/p>\r\n<p class=\"Example\">If we created 100 confidence intervals of the same size from the same population, we would expect 95 of them to contain the true parameter (the population mean weight). We also expect five of the intervals would not contain the parameter.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"656\"]<img class=\"frame-47\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170647\/3247.png\" alt=\"3247.png\" width=\"656\" height=\"583\" \/> Figure 1. Confidence intervals from twenty-five different samples.[\/caption]\r\n<p class=\"Example\">In this example, twenty-five samples from the same population gave these 95% confidence intervals. In the long term, 95% of all samples give an interval that contains \u00b5, the true (but unknown) population mean.<\/p>\r\n\r\n<\/div>\r\nLevel of confidence is expressed as a percent.\r\n<ul>\r\n \t<li class=\"Bullet-List\">The compliment to the level of confidence is <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> (alpha), the level of significance.<\/li>\r\n \t<li class=\"Bullet-List\">The level of confidence is described as (1- <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>) * 100%.<\/li>\r\n<\/ul>\r\nWhat does this really mean?\r\n<ul>\r\n \t<li class=\"Bullet-List\">We use a point estimate (e.g., sample mean) to estimate the population mean.<\/li>\r\n \t<li class=\"Bullet-List\">We attach a level of confidence to this interval to describe how certain we are that this interval actually contains the unknown population parameter.<\/li>\r\n \t<li class=\"Bullet-List\">We want to estimate the population parameter, such as the mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>) or proportion (<em>p<\/em>).<\/li>\r\n<\/ul>\r\n<p class=\"ExampleCenter\" style=\"text-align: center\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170649\/3239.png\" alt=\"3239.png\" \/><\/span>&lt; <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span> &lt; <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170649\/3231.png\" alt=\"3231.png\" \/>\u00a0<\/span>or \u00a0<span class=\"Inline-Equation\"><img class=\"frame-42\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170650\/3220.png\" alt=\"3220.png\" \/><\/span>&lt; p &lt; <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170651\/3208.png\" alt=\"3208.png\" \/><\/span><\/p>\r\nwhere <em>E<\/em> is the margin of error.\r\n\r\nThe confidence is based on area under a normal curve. So the assumption of normality must be met (see\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/chapter\/chapter-1-descriptive-statistics-and-the-normal-distribution\/\">Chapter 1<\/a>).\r\n<h3>Confidence Intervals about the Mean (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong>) when the Population Standard Deviation (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong>) is Known<\/h3>\r\nA confidence interval takes the form of: <strong class=\"Strong-2\">point estimate \u00b1 margin of error.<\/strong>\r\n<h5>The point estimate<\/h5>\r\n<ul>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">The point estimate comes from the sample data.<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">To estimate the population mean (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong>), use the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) as the point estimate.<\/li>\r\n<\/ul>\r\n<h5>The margin of error<\/h5>\r\n<ul>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">Depends on the level of confidence, the sample size and the population standard deviation.<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">It is computed as <img class=\"frame-45\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170652\/3192.png\" alt=\"3192.png\" width=\"87\" height=\"42\" \/>where <img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170652\/3185.png\" alt=\"3185.png\" width=\"31\" height=\"26\" \/>is the critical value from the standard normal table associated with <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> (the level of significance).<\/li>\r\n<\/ul>\r\n<h5>The critical value <span class=\"Inline-Equation-Large\"><img class=\"frame-20\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170653\/3177.png\" alt=\"3177.png\" width=\"31\" height=\"26\" \/><\/span><\/h5>\r\n<ul>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">This is a Z-score that bounds the level of confidence.<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">Confidence intervals are ALWAYS two-sided and the Z-scores are the limits of the area associated with the level of confidence.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"451\"]<img class=\"frame-49\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170654\/3168.png\" alt=\"3168.png\" width=\"451\" height=\"347\" \/> Figure 2. The middle 95% area under a standard normal curve.[\/caption]\r\n<ul>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">The level of significance (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>) is divided into halves because we are looking at the middle 95% of the area under the curve.<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">Go to your standard normal table and find the area of 0.025 in the body of values.<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">What is the Z-score for that area?<\/li>\r\n \t<li class=\"List-Paragraph-Bullet-level-2\">The Z-scores of \u00b1 1.96 are the critical Z-scores for a 95% confidence interval.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"685\"]<img class=\"frame-50\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170656\/3159.png\" alt=\"3159.png\" width=\"685\" height=\"223\" \/> Table 1. Common critical values (Z-scores).[\/caption]\r\n\r\nConstruction of a confidence interval about <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong> when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is known:\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170657\/3151.png\" alt=\"3151.png\" width=\"33\" height=\"36\" \/><\/span> (critical value)<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170658\/3144.png\" alt=\"3144.png\" width=\"115\" height=\"56\" \/><\/span> (margin of error)<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170659\/3136.png\" alt=\"3136.png\" width=\"52\" height=\"25\" \/><\/span> (point estimate \u00b1 margin of error)<\/li>\r\n<\/ol>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\n<p class=\"Example\">Construct a confidence interval about the population mean.<\/p>\r\n<p class=\"Example\">Researchers have been studying p-loading in Jones Lake for many years. It is known that mean water clarity (using a Secchi disk) is normally distributed with a population standard deviation of <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> = 15.4 in. A random sample of 22 measurements was taken at various points on the lake with a sample mean of <span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span> = 57.8 in. The researchers want you to construct a 95% confidence interval for <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>, the mean water clarity.<\/p>\r\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170659\/3114.png\" alt=\"3114.png\" width=\"36\" height=\"39\" \/><\/span> = 1.96<\/p>\r\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170700\/3105.png\" alt=\"3105.png\" width=\"113\" height=\"55\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img class=\"frame-58\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170701\/3098.png\" alt=\"3098.png\" width=\"159\" height=\"56\" \/><\/span><\/p>\r\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170702\/3091.png\" alt=\"3091.png\" width=\"51\" height=\"24\" \/><\/span> = 57.8 \u00b1 6.435<\/p>\r\n<p class=\"ExampleCenter\">95% confidence interval for the mean water clarity is (51.36, 64.24).<\/p>\r\n<p class=\"ExampleCenter\">We can be 95% confident that this interval contains the population mean water clarity for Jones Lake.<\/p>\r\n<p class=\"Example\">Now construct a 99% confidence interval for <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>, the mean water clarity, and interpret.<\/p>\r\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170702\/3083.png\" alt=\"3083.png\" width=\"37\" height=\"41\" \/><\/span>= 2.575<\/p>\r\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170703\/3074.png\" alt=\"3074.png\" width=\"121\" height=\"59\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img class=\"frame-51\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170704\/3065.png\" alt=\"3065.png\" width=\"187\" height=\"61\" \/><\/span><\/p>\r\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170705\/3058.png\" alt=\"3058.png\" width=\"55\" height=\"26\" \/><\/span> = 57.8\u00b1 8.454<\/p>\r\n<p class=\"Example\">99% confidence interval for the mean water clarity is (49.35, 66.25).<\/p>\r\n<p class=\"Example\">We can be 99% confident that this interval contains the population mean water clarity for Jones Lake.<\/p>\r\n<p class=\"Example\">As the level of confidence increased from 95% to 99%, the width of the interval increased. As the probability (area under the normal curve) increased, the critical value increased resulting in a wider interval.<\/p>\r\n\r\n<\/div>\r\n<h2>Software Solutions<\/h2>\r\n<h3>Minitab<\/h3>\r\n<p class=\"Example\">You can use Minitab to construct this 95% confidence interval (Excel does not construct confidence intervals about the mean when the population standard deviation is known). Select Basic Statistics&gt;1-sample Z. Enter the known population standard deviation and select the required level of confidence.<\/p>\r\n<img class=\"frame-19 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170707\/030_2_fmt.png\" alt=\"030_2.tif\" width=\"423\" height=\"643\" \/>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"390\"]<img class=\"frame-52\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170710\/030_1_fmt.png\" alt=\"030_1.tif\" width=\"390\" height=\"643\" \/> Figure 3. Minitab screen shots for constructing a confidence interval.[\/caption]\r\n<h4>One-Sample Z: depth<\/h4>\r\n<table class=\"Table\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"Table\" colspan=\"6\">\r\n<p class=\"Example\">The assumed standard deviation = 15.4<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"Table\">\r\n<p class=\"Table\">Variable<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">N<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">Mean<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">StDev<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">SE Mean<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">95% CI<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"Table\">\r\n<p class=\"Table\">depth<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">22<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">57.80<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">11.60<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">3.28<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">(51.36, 64.24)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Confidence Intervals about the Mean (<strong class=\"char-style-override-2\">\u03bc<\/strong>) when the Population Standard Deviation (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong>) is Unknown<\/h3>\r\nTypically, in real life we often don\u2019t know the population standard deviation (\u03c3). We can use the sample standard deviation (<em>s<\/em>) in place of \u03c3. However, because of this change, we can\u2019t use the standard normal distribution to find the critical values necessary for constructing a confidence interval.\r\n\r\nThe Student\u2019s t-distribution was created for situations when \u03c3 was unknown. Gosset worked as a quality control engineer for Guinness Brewery in Dublin. He found errors in his testing and he knew it was due to the use of s instead of \u03c3. He created this distribution to deal with the problem of an unknown population standard deviation and small sample sizes. A portion of the t-table is shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"857\"]<img class=\"frame-13\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170712\/3032.png\" alt=\"3032.png\" width=\"857\" height=\"194\" \/> Table 2. Portion of the student\u2019s t-table.[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\n<p class=\"ExampleHeading\">Find the critical value <span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170714\/3023.png\" alt=\"3023.png\" width=\"23\" height=\"32\" \/><\/span> for a 95% confidence interval with a sample size of n=13.<\/p>\r\n\r\n<ul>\r\n \t<li class=\"ExampleList\">Degrees of freedom (down the left-hand column) is equal to n-1 = 12<\/li>\r\n \t<li class=\"ExampleList\"><span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> = 0.05 and <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>\/2 = 0.025<\/li>\r\n \t<li class=\"ExampleList\">Go down the 0.025 column to 12 df<\/li>\r\n \t<li class=\"ExampleList\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170715\/3013.png\" alt=\"3013.png\" width=\"24\" height=\"34\" \/>= 2.179<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe critical values from the students\u2019 t-distribution approach the critical values from the standard normal distribution as the sample size (n) increases.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"498\"]<img class=\"frame-57\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170716\/3002.png\" alt=\"3002.png\" width=\"498\" height=\"201\" \/> Table 3. Critical values from the student\u2019s t-table.[\/caption]\r\n\r\nUsing the standard normal curve, the critical value for a 95% confidence interval is <strong class=\"Strong-2\">1.96<\/strong>. You can see how different samples sizes will change the critical value and thus the confidence interval, especially when the sample size is small.\r\n<h3>Construction of a Confidence Interval about <strong class=\"char-style-override-2\">\u03bc<\/strong> when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Unknown<\/h3>\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170718\/2995.png\" alt=\"2995.png\" width=\"28\" height=\"39\" \/><\/span> critical value with n-1 df<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170719\/2987.png\" alt=\"2987.png\" width=\"103\" height=\"53\" \/><\/span><\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170720\/2979.png\" alt=\"2979.png\" width=\"50\" height=\"24\" \/><\/span><\/li>\r\n<\/ol>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\n<p class=\"Example\">Researchers studying the effects of acid rain in the Adirondack Mountains collected water samples from 22 lakes. They measured the pH (acidity) of the water and want to construct a 99% confidence interval about the mean lake pH for this region. The sample mean is 6.4438 with a sample standard deviation of 0.7120. They do not know anything about the distribution of the pH of this population, and the sample is small (n&lt;30), so they look at a normal probability plot.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"901\"]<img class=\"frame-13\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170721\/2970.png\" alt=\"2970.png\" width=\"901\" height=\"600\" \/> Figure 4. Normal probability plot.[\/caption]\r\n<p class=\"Example\">The data is normally distributed. Now construct the 99% confidence interval about the mean pH.<\/p>\r\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170723\/2963.png\" alt=\"2963.png\" width=\"29\" height=\"41\" \/><\/span> = 2.831<\/p>\r\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170724\/2955.png\" alt=\"2955.png\" width=\"115\" height=\"59\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170725\/2948.png\" alt=\"2948.png\" width=\"120\" height=\"54\" \/><\/span>= 0.4297<\/p>\r\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170726\/2940.png\" alt=\"2940.png\" width=\"53\" height=\"25\" \/><\/span> = 6.443 \u00b1 0.4297<\/p>\r\n<p class=\"Example\">The 99% confidence interval about the mean pH is (6.013, 6.863).<\/p>\r\n<p class=\"Example\">We are 99% confident that this interval contains the mean lake pH for this lake population.<\/p>\r\n<p class=\"Example\">Now construct a 90% confidence interval about the mean pH for these lakes.<\/p>\r\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170726\/2929.png\" alt=\"2929.png\" width=\"30\" height=\"42\" \/><\/span> = 1.721<\/p>\r\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170727\/2919.png\" alt=\"2919.png\" width=\"117\" height=\"60\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170728\/2909.png\" alt=\"2909.png\" width=\"119\" height=\"55\" \/><\/span>= 0.2612<\/p>\r\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170729\/2900.png\" alt=\"2900.png\" width=\"55\" height=\"26\" \/><\/span> = 6.443 \u00b1 0.2612<\/p>\r\n<p class=\"Example\">The 90% confidence interval about the mean pH is (6.182, 6.704).<\/p>\r\n<p class=\"Example\">We are 90% confident that this interval contains the mean lake pH for this lake population.<\/p>\r\n<p class=\"Example\">Notice how the width of the interval decreased as the level of confidence decreased from 99 to 90%.<\/p>\r\n<p class=\"Example\">Construct a 90% confidence interval about the mean lake pH using Excel and Minitab.<\/p>\r\n\r\n<\/div>\r\n<h2>Software Solutions<\/h2>\r\n<h3>Minitab<\/h3>\r\n<p class=\"Example\">For Minitab, enter the data in the spreadsheet and select Basic statistics and 1-sample t-test.<\/p>\r\n<p class=\"No-Caption\"><img class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170731\/035_2_fmt.png\" alt=\"035_2.tif\" \/><img class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170734\/035_1_fmt.png\" alt=\"035_1.tif\" \/><\/p>\r\n\r\n<h4>One-Sample T: pH<\/h4>\r\n<table class=\"Table\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 43px\">\r\n<td class=\"Table\" style=\"height: 43px\">Variable<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">N<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">Mean<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">StDev<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">SE Mean<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">90% CI<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43.1875px\">\r\n<td class=\"Table\" style=\"height: 43.1875px\">pH<\/td>\r\n<td class=\"Table\" style=\"height: 43.1875px\">\r\n<p class=\"Table\">22<\/p>\r\n<\/td>\r\n<td class=\"Table\" style=\"height: 43.1875px\">\r\n<p class=\"Table\">6.443<\/p>\r\n<\/td>\r\n<td class=\"Table\" style=\"height: 43.1875px\">\r\n<p class=\"Table\">0.712<\/p>\r\n<\/td>\r\n<td class=\"Table\" style=\"height: 43.1875px\">\r\n<p class=\"Table\">0.152<\/p>\r\n<\/td>\r\n<td class=\"Table\" style=\"height: 43.1875px\">\r\n<p class=\"Table\">(6.182, 6.704)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"Example\">Additional example:<\/p>\r\nhttps:\/\/youtu.be\/gIyPlEJE6Jc\r\n<h3>For Excel, enter the data in the spreadsheet and select descriptive statistics. Check Summary Statistics and select the level and confidence.Excel<\/h3>\r\n<p class=\"No-Caption\"><span class=\"Equation-Right\"><img class=\"frame-52\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170737\/034_2_fmt.png\" alt=\"034_2.tif\" \/><\/span><span class=\"Equation-Left\"><img class=\"frame-29\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170740\/034_1_fmt.png\" alt=\"034_1.tif\" \/><\/span><\/p>\r\n\r\n<table id=\"table-3\" class=\"Table\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Mean<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">6.442909<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Standard Error<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">0.151801<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Median<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">6.4925<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Mode<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">#N\/A<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Standard Deviation<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">0.712008<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Sample Variance<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">0.506956<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Kurtosis<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">-0.5007<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Skewness<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">-0.60591<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Range<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">2.338<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Minimum<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">5.113<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Maximum<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">7.451<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Sum<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">141.744<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p class=\"Table\">Count<\/p>\r\n<\/td>\r\n<td>\r\n<p class=\"Table\">22<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"Table\">\r\n<p class=\"Table\">Confidence Level(90.0%)<\/p>\r\n<\/td>\r\n<td class=\"Table\">\r\n<p class=\"Table\">0.26121<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"Example\">Excel gives you the sample mean in the first line (6.442909) and the margin of error in the last line (0.26121). You must complete the computation yourself to obtain the interval (6.442909\u00b10.26121).<\/p>\r\n\r\n<h3>Confidence Intervals about the Population Proportion (<em class=\"char-style-override-2\">p<\/em>)<\/h3>\r\nFrequently, we are interested in estimating the population proportion (<span class=\"BoldItalic Strong-2\">p<\/span>), instead of the population mean (<strong class=\"Strong-2\">\u00b5<\/strong>). For example, you may need to estimate the proportion of trees infected with beech bark disease, or the proportion of people who support \u201cgreen\u201d products. The parameter <span class=\"BoldItalic Strong-2\">p<\/span> can be estimated in the same ways as we estimated <strong class=\"Strong-2\">\u00b5<\/strong>, the population mean.\r\n<h4>The Sample Proportion<\/h4>\r\n<ul>\r\n \t<li class=\"List-Paragraph\">The sample proportion is the best point estimate for the true population proportion.<\/li>\r\n \t<li class=\"List-Paragraph\">Sample proportion\u00a0<img class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170741\/2862.png\" alt=\"2862.png\" width=\"40\" height=\"41\" \/>\u00a0where <em>x<\/em> is the number of elements in the sample with the characteristic you are interested in, and <em>n<\/em> is the sample size.<\/li>\r\n<\/ul>\r\n<h4>The Assumption of Normality when Estimating Proportions<\/h4>\r\n<ul>\r\n \t<li class=\"List-Paragraph\">The assumption of a normally distributed population is still important, even though the parameter has changed.<\/li>\r\n \t<li class=\"List-Paragraph\">Normality can be verified if:<\/li>\r\n \t<li>\r\n<p class=\"Centered\"><span class=\"Picture\"><img class=\"frame-58 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170742\/2856.png\" alt=\"2856.png\" \/><\/span><\/p>\r\n<\/li>\r\n<\/ul>\r\n<h4>Constructing a Confidence Interval about the Population Proportion<\/h4>\r\nConstructing a confidence interval about the proportion follows the same three steps we have used in previous examples.\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170743\/2848.png\" alt=\"2848.png\" width=\"32\" height=\"35\" \/>(critical value from the standard normal table)<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><img class=\"frame-26\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170744\/2842.png\" alt=\"2842.png\" width=\"164\" height=\"59\" \/>(margin of error)<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-18\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170744\/2836.png\" alt=\"2836.png\" width=\"54\" height=\"28\" \/><\/span> (point estimate \u00b1 margin of error)<\/li>\r\n<\/ol>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\n<p class=\"ExampleHeading\">A botanist has produced a new variety of hybrid soybean that is better able to withstand drought. She wants to construct a 95% confidence interval about the germination rate (percent germination). She randomly selected 500 seeds and found that 421 have germinated.<\/p>\r\n<p class=\"Example\">First, compute the point estimate<\/p>\r\n<p class=\"ExampleCenter\"><span class=\"Picture\"><img class=\"frame-7 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170745\/2827.png\" alt=\"2827.png\" width=\"190\" height=\"58\" \/><\/span><\/p>\r\n<p class=\"Example\">Check normality:\u00a0<span class=\"Inline-Equation\"><img class=\"frame-53\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170746\/2818.png\" alt=\"2818.png\" width=\"415\" height=\"26\" \/><\/span><\/p>\r\n<p class=\"Example\">You can assume a normal distribution.<\/p>\r\n<p class=\"Example\">Now construct the confidence interval:<\/p>\r\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img class=\"frame-20\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170748\/2811.png\" alt=\"2811.png\" width=\"28\" height=\"30\" \/><\/span> = 1.96<\/p>\r\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img class=\"frame-54\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170748\/2802.png\" alt=\"2802.png\" width=\"174\" height=\"55\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img class=\"frame-55\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170749\/2794.png\" alt=\"2794.png\" width=\"227\" height=\"57\" \/><\/span><\/p>\r\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img class=\"frame-56\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170750\/2784.png\" alt=\"2784.png\" width=\"163\" height=\"21\" \/><\/span><\/p>\r\n<p class=\"Example\">The 95% confidence interval for the germination rate is (81.0%, 87.4%).<\/p>\r\n<p class=\"Example\">We can be 95% confident that this interval contains the true germination rate for this population.<\/p>\r\n\r\n<\/div>\r\n<h2>Software Solutions<\/h2>\r\n<h3>Minitab<\/h3>\r\n<p class=\"Example\">You can use Minitab to compute the confidence interval. Select STAT&gt;Basic stats&gt;1-proportion. Select summarized data and enter the number of events (421) and the number of trials (500). Click Options and select the correct confidence level. Check \u201ctest and interval based on normal distribution\u201d if the assumption of normality has been verified.<\/p>\r\n<p class=\"Centered\"><img class=\"frame-57\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170752\/036_1_fmt.png\" alt=\"036_1.tif\" \/>\u00a0<img class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170755\/036_2_fmt.png\" alt=\"036_2.tif\" \/><\/p>\r\n\r\n<h4>Test and CI for One Proportion<\/h4>\r\n<table class=\"Table\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 43.6562px\">\r\n<td class=\"Table\" style=\"height: 43.6562px\">Sample<\/td>\r\n<td class=\"Table\" style=\"height: 43.6562px\">X<\/td>\r\n<td class=\"Table\" style=\"height: 43.6562px\">N<\/td>\r\n<td class=\"Table\" style=\"height: 43.6562px\">Sample p<\/td>\r\n<td class=\"Table\" style=\"height: 43.6562px\">95% CI<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43px\">\r\n<td class=\"Table\" style=\"height: 43px\">1<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">421<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">500<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">0.842000<\/td>\r\n<td class=\"Table\" style=\"height: 43px\">(0.810030, 0.873970)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"Example\">Using the normal approximation.<\/p>\r\n\r\n<h3>Excel<\/h3>\r\n<p class=\"Example\">Excel does not compute confidence intervals for estimating the population proportion.<\/p>\r\n\r\n<h3>Confidence Interval Summary<\/h3>\r\nWhich method do I use?\r\n\r\nThe first question to ask yourself is: <strong class=\"Strong-2\">Which parameter are you trying to estimate<\/strong><strong>?<\/strong> If it is the mean (<em>\u00b5<\/em>), then ask yourself: <strong class=\"Strong-2\">Is the population standard deviation (<\/strong><strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong><strong class=\"Strong-2\">) known<\/strong><strong>?<\/strong> If yes, then follow the next 3 steps:\r\n<h4>Confidence Interval about the Population Mean (\u00b5) when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Known<\/h4>\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170757\/2755.png\" alt=\"2755.png\" width=\"35\" height=\"38\" \/><\/span> critical value (from the standard normal table)<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170757\/2747.png\" alt=\"2747.png\" width=\"115\" height=\"56\" \/><\/span><\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170758\/2739.png\" alt=\"2739.png\" width=\"51\" height=\"24\" \/><\/span><\/li>\r\n<\/ol>\r\nIf no, follow these 3 steps:\r\n<h4>Confidence Interval about the Population Mean (\u00b5) when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Unknown<\/h4>\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170759\/2731.png\" alt=\"2731.png\" width=\"28\" height=\"40\" \/> critical value with n-1 df from the student t-distribution<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170759\/2723.png\" alt=\"2723.png\" width=\"115\" height=\"59\" \/><\/span><\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170800\/2714.png\" alt=\"2714.png\" width=\"53\" height=\"25\" \/><\/span><\/li>\r\n<\/ol>\r\nIf you want to construct a confidence interval about the population proportion, follow these 3 steps:\r\n<h4>Confidence Interval about the Proportion<\/h4>\r\n<ol>\r\n \t<li class=\"List-Paragraph-Number-1\"><img class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170801\/2707.png\" alt=\"2707.png\" width=\"38\" height=\"41\" \/> critical value from the standard normal table<\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img class=\"frame-58\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170801\/2696.png\" alt=\"2696.png\" width=\"173\" height=\"58\" \/><\/span><\/li>\r\n \t<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170802\/2685.png\" alt=\"2685.png\" width=\"52\" height=\"25\" \/><\/span><\/li>\r\n<\/ol>\r\nRemember that the assumption of normality must be verified.\r\n\r\n<\/div>","rendered":"<div class=\"Basic-Text-Frame\">\n<h2 class=\"Chapter-Number\">Sampling Distribution of the Sample Mean<\/h2>\n<p>Inferential testing uses the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) to estimate the population mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>). Typically, we use the data from a single sample, but there are many possible samples of the same size that could be drawn from that population. As we saw in the previous chapter, the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) is a random variable with its own distribution.<\/p>\n<ul>\n<li class=\"List-Paragraph\">The distribution of the sample mean will have a mean equal to \u00b5.<\/li>\n<li class=\"List-Paragraph\">It will have a standard deviation (standard error) equal to <img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170640\/3341.png\" alt=\"3341.png\" width=\"35\" height=\"32\" \/><\/li>\n<\/ul>\n<p>Because our inferences about the population mean rely on the sample mean, we focus on the distribution of the sample mean. Is it normal? What if our population is not normally distributed or we don\u2019t know anything about the distribution of our population?<\/p>\n<p><strong>The Central Limit Theorem<\/strong> states that the sampling distribution of the sample means will approach a normal distribution as the sample size increases.<\/p>\n<ul>\n<li class=\"List-Paragraph\">So if we do not have a normal distribution, or know nothing about our distribution, the CLT tells us that the distribution of the sample means (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) will become normal distributed as <em>n<\/em> (sample size) increases.<\/li>\n<li class=\"List-Paragraph\">How large does <em>n<\/em> have to be?<\/li>\n<li class=\"List-Paragraph\">A general rule of thumb tells us that <em>n<\/em> \u2265 30.<\/li>\n<\/ul>\n<p>The Central Limit Theorem tells us that regardless of the shape of our population, the sampling distribution of the sample mean will be normal as the sample size increases.<\/p>\n<h3>Sampling Distribution of the Sample Proportion<\/h3>\n<p>The population proportion (<em>p<\/em>) is a parameter that is as commonly estimated as the mean. It is just as important to understand the distribution of the sample proportion, as the mean. With proportions, the element either has the characteristic you are interested in or the element does not have the characteristic. The sample proportion (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>p\u0302<\/em><\/span>) is calculated by<\/p>\n<p class=\"Centered\"><span class=\"Inline-Equation-Large\"><img decoding=\"async\" class=\"frame-43 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170641\/3316.png\" alt=\"3316.png\" \/><\/span><\/p>\n<p>where <em>x<\/em> is the number of elements in your population with the characteristic and <em>n<\/em> is the sample size.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p class=\"Example\">You are studying the number of cavity trees in the Monongahela National Forest for wildlife habitat. You have a sample size of n = 950 trees and, of those trees, x = 238 trees with cavities. The sample proportion is:<\/p>\n<p class=\"ExampleCenter\"><img decoding=\"async\" class=\"frame-44\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170642\/3305.png\" alt=\"3305.png\" \/><\/p>\n<\/div>\n<p>The distribution of the sample proportion has a mean of <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170643\/3295.png\" alt=\"3295.png\" width=\"55\" height=\"21\" \/><\/span><\/p>\n<p>and has a standard deviation of <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-45\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170644\/3287.png\" alt=\"3287.png\" width=\"121\" height=\"50\" \/><\/span> .<\/p>\n<p>The sample proportion is normally distributed if <em>n<\/em> is very large and <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-5\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170645\/3282.png\" alt=\"3282.png\" width=\"10\" height=\"19\" \/><\/span> isn\u2019t close to 0 or 1. We can also use the following relationship to assess normality when the parameter being estimated is <em>p<\/em>, the population proportion:<\/p>\n<p class=\"Centered\"><img decoding=\"async\" class=\"frame-46 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170646\/3273.png\" alt=\"3273.png\" \/><\/p>\n<h2>Confidence Intervals<\/h2>\n<p>In the preceding chapter we learned that populations are characterized by descriptive measures called parameters. Inferences about parameters are based on sample statistics. We now want to estimate population parameters and assess the reliability of our estimates based on our knowledge of the sampling distributions of these statistics.<\/p>\n<h3>Point Estimates<\/h3>\n<p>We start with a point estimate. This is a single value computed from the sample data that is used to estimate the population parameter of interest.<\/p>\n<ul>\n<li class=\"List-Paragraph\">The sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) is a point estimate of the population mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>).<\/li>\n<li class=\"List-Paragraph\">The sample proportion (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>p\u0302<\/em><\/span>) is the point estimate of the population proportion (<em>p<\/em>).<\/li>\n<\/ul>\n<p>We use point estimates to construct confidence intervals for unknown parameters.<\/p>\n<ul>\n<li class=\"List-Paragraph\">A confidence interval is an interval of values instead of a single point estimate.<\/li>\n<li class=\"List-Paragraph\">The level of confidence corresponds to the expected proportion of intervals that will contain the parameter if many confidence intervals are constructed of the same sample size from the same population.<\/li>\n<li class=\"List-Paragraph\">Our uncertainty is about whether our particular confidence interval is one of those that truly contains the true value of the parameter.<\/li>\n<\/ul>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p class=\"Example\">We are 95% confident that our interval contains the population mean bear weight.<\/p>\n<p class=\"Example\">If we created 100 confidence intervals of the same size from the same population, we would expect 95 of them to contain the true parameter (the population mean weight). We also expect five of the intervals would not contain the parameter.<\/p>\n<div style=\"width: 666px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-47\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170647\/3247.png\" alt=\"3247.png\" width=\"656\" height=\"583\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Confidence intervals from twenty-five different samples.<\/p>\n<\/div>\n<p class=\"Example\">In this example, twenty-five samples from the same population gave these 95% confidence intervals. In the long term, 95% of all samples give an interval that contains \u00b5, the true (but unknown) population mean.<\/p>\n<\/div>\n<p>Level of confidence is expressed as a percent.<\/p>\n<ul>\n<li class=\"Bullet-List\">The compliment to the level of confidence is <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> (alpha), the level of significance.<\/li>\n<li class=\"Bullet-List\">The level of confidence is described as (1- <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>) * 100%.<\/li>\n<\/ul>\n<p>What does this really mean?<\/p>\n<ul>\n<li class=\"Bullet-List\">We use a point estimate (e.g., sample mean) to estimate the population mean.<\/li>\n<li class=\"Bullet-List\">We attach a level of confidence to this interval to describe how certain we are that this interval actually contains the unknown population parameter.<\/li>\n<li class=\"Bullet-List\">We want to estimate the population parameter, such as the mean (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>) or proportion (<em>p<\/em>).<\/li>\n<\/ul>\n<p class=\"ExampleCenter\" style=\"text-align: center\"><span class=\"Inline-Equation\"><img decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170649\/3239.png\" alt=\"3239.png\" \/><\/span>&lt; <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span> &lt; <span class=\"Inline-Equation\"><img decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170649\/3231.png\" alt=\"3231.png\" \/>\u00a0<\/span>or \u00a0<span class=\"Inline-Equation\"><img decoding=\"async\" class=\"frame-42\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170650\/3220.png\" alt=\"3220.png\" \/><\/span>&lt; p &lt; <span class=\"Inline-Equation\"><img decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170651\/3208.png\" alt=\"3208.png\" \/><\/span><\/p>\n<p>where <em>E<\/em> is the margin of error.<\/p>\n<p>The confidence is based on area under a normal curve. So the assumption of normality must be met (see\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/chapter\/chapter-1-descriptive-statistics-and-the-normal-distribution\/\">Chapter 1<\/a>).<\/p>\n<h3>Confidence Intervals about the Mean (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong>) when the Population Standard Deviation (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong>) is Known<\/h3>\n<p>A confidence interval takes the form of: <strong class=\"Strong-2\">point estimate \u00b1 margin of error.<\/strong><\/p>\n<h5>The point estimate<\/h5>\n<ul>\n<li class=\"List-Paragraph-Bullet-level-2\">The point estimate comes from the sample data.<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">To estimate the population mean (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong>), use the sample mean (<span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span>) as the point estimate.<\/li>\n<\/ul>\n<h5>The margin of error<\/h5>\n<ul>\n<li class=\"List-Paragraph-Bullet-level-2\">Depends on the level of confidence, the sample size and the population standard deviation.<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">It is computed as <img loading=\"lazy\" decoding=\"async\" class=\"frame-45\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170652\/3192.png\" alt=\"3192.png\" width=\"87\" height=\"42\" \/>where <img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170652\/3185.png\" alt=\"3185.png\" width=\"31\" height=\"26\" \/>is the critical value from the standard normal table associated with <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> (the level of significance).<\/li>\n<\/ul>\n<h5>The critical value <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-20\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170653\/3177.png\" alt=\"3177.png\" width=\"31\" height=\"26\" \/><\/span><\/h5>\n<ul>\n<li class=\"List-Paragraph-Bullet-level-2\">This is a Z-score that bounds the level of confidence.<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">Confidence intervals are ALWAYS two-sided and the Z-scores are the limits of the area associated with the level of confidence.<\/li>\n<\/ul>\n<div style=\"width: 461px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-49\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170654\/3168.png\" alt=\"3168.png\" width=\"451\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The middle 95% area under a standard normal curve.<\/p>\n<\/div>\n<ul>\n<li class=\"List-Paragraph-Bullet-level-2\">The level of significance (<span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>) is divided into halves because we are looking at the middle 95% of the area under the curve.<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">Go to your standard normal table and find the area of 0.025 in the body of values.<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">What is the Z-score for that area?<\/li>\n<li class=\"List-Paragraph-Bullet-level-2\">The Z-scores of \u00b1 1.96 are the critical Z-scores for a 95% confidence interval.<\/li>\n<\/ul>\n<div style=\"width: 695px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-50\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170656\/3159.png\" alt=\"3159.png\" width=\"685\" height=\"223\" \/><\/p>\n<p class=\"wp-caption-text\">Table 1. Common critical values (Z-scores).<\/p>\n<\/div>\n<p>Construction of a confidence interval about <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03bc<\/strong> when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is known:<\/p>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170657\/3151.png\" alt=\"3151.png\" width=\"33\" height=\"36\" \/><\/span> (critical value)<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170658\/3144.png\" alt=\"3144.png\" width=\"115\" height=\"56\" \/><\/span> (margin of error)<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170659\/3136.png\" alt=\"3136.png\" width=\"52\" height=\"25\" \/><\/span> (point estimate \u00b1 margin of error)<\/li>\n<\/ol>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p class=\"Example\">Construct a confidence interval about the population mean.<\/p>\n<p class=\"Example\">Researchers have been studying p-loading in Jones Lake for many years. It is known that mean water clarity (using a Secchi disk) is normally distributed with a population standard deviation of <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> = 15.4 in. A random sample of 22 measurements was taken at various points on the lake with a sample mean of <span class=\"Symbols\" xml:lang=\"ar-SA\"><em>x\u0304<\/em><\/span> = 57.8 in. The researchers want you to construct a 95% confidence interval for <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>, the mean water clarity.<\/p>\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170659\/3114.png\" alt=\"3114.png\" width=\"36\" height=\"39\" \/><\/span> = 1.96<\/p>\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170700\/3105.png\" alt=\"3105.png\" width=\"113\" height=\"55\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-58\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170701\/3098.png\" alt=\"3098.png\" width=\"159\" height=\"56\" \/><\/span><\/p>\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170702\/3091.png\" alt=\"3091.png\" width=\"51\" height=\"24\" \/><\/span> = 57.8 \u00b1 6.435<\/p>\n<p class=\"ExampleCenter\">95% confidence interval for the mean water clarity is (51.36, 64.24).<\/p>\n<p class=\"ExampleCenter\">We can be 95% confident that this interval contains the population mean water clarity for Jones Lake.<\/p>\n<p class=\"Example\">Now construct a 99% confidence interval for <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03bc<\/span>, the mean water clarity, and interpret.<\/p>\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170702\/3083.png\" alt=\"3083.png\" width=\"37\" height=\"41\" \/><\/span>= 2.575<\/p>\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170703\/3074.png\" alt=\"3074.png\" width=\"121\" height=\"59\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-51\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170704\/3065.png\" alt=\"3065.png\" width=\"187\" height=\"61\" \/><\/span><\/p>\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170705\/3058.png\" alt=\"3058.png\" width=\"55\" height=\"26\" \/><\/span> = 57.8\u00b1 8.454<\/p>\n<p class=\"Example\">99% confidence interval for the mean water clarity is (49.35, 66.25).<\/p>\n<p class=\"Example\">We can be 99% confident that this interval contains the population mean water clarity for Jones Lake.<\/p>\n<p class=\"Example\">As the level of confidence increased from 95% to 99%, the width of the interval increased. As the probability (area under the normal curve) increased, the critical value increased resulting in a wider interval.<\/p>\n<\/div>\n<h2>Software Solutions<\/h2>\n<h3>Minitab<\/h3>\n<p class=\"Example\">You can use Minitab to construct this 95% confidence interval (Excel does not construct confidence intervals about the mean when the population standard deviation is known). Select Basic Statistics&gt;1-sample Z. Enter the known population standard deviation and select the required level of confidence.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"frame-19 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170707\/030_2_fmt.png\" alt=\"030_2.tif\" width=\"423\" height=\"643\" \/><\/p>\n<div style=\"width: 400px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-52\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170710\/030_1_fmt.png\" alt=\"030_1.tif\" width=\"390\" height=\"643\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Minitab screen shots for constructing a confidence interval.<\/p>\n<\/div>\n<h4>One-Sample Z: depth<\/h4>\n<table class=\"Table\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"Table\" colspan=\"6\">\n<p class=\"Example\">The assumed standard deviation = 15.4<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"Table\">\n<p class=\"Table\">Variable<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">N<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">Mean<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">StDev<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">SE Mean<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">95% CI<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"Table\">\n<p class=\"Table\">depth<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">22<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">57.80<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">11.60<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">3.28<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">(51.36, 64.24)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Confidence Intervals about the Mean (<strong class=\"char-style-override-2\">\u03bc<\/strong>) when the Population Standard Deviation (<strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong>) is Unknown<\/h3>\n<p>Typically, in real life we often don\u2019t know the population standard deviation (\u03c3). We can use the sample standard deviation (<em>s<\/em>) in place of \u03c3. However, because of this change, we can\u2019t use the standard normal distribution to find the critical values necessary for constructing a confidence interval.<\/p>\n<p>The Student\u2019s t-distribution was created for situations when \u03c3 was unknown. Gosset worked as a quality control engineer for Guinness Brewery in Dublin. He found errors in his testing and he knew it was due to the use of s instead of \u03c3. He created this distribution to deal with the problem of an unknown population standard deviation and small sample sizes. A portion of the t-table is shown below.<\/p>\n<div style=\"width: 867px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-13\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170712\/3032.png\" alt=\"3032.png\" width=\"857\" height=\"194\" \/><\/p>\n<p class=\"wp-caption-text\">Table 2. Portion of the student\u2019s t-table.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p class=\"ExampleHeading\">Find the critical value <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170714\/3023.png\" alt=\"3023.png\" width=\"23\" height=\"32\" \/><\/span> for a 95% confidence interval with a sample size of n=13.<\/p>\n<ul>\n<li class=\"ExampleList\">Degrees of freedom (down the left-hand column) is equal to n-1 = 12<\/li>\n<li class=\"ExampleList\"><span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span> = 0.05 and <span class=\"Symbols\" xml:lang=\"ar-SA\">\u03b1<\/span>\/2 = 0.025<\/li>\n<li class=\"ExampleList\">Go down the 0.025 column to 12 df<\/li>\n<li class=\"ExampleList\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170715\/3013.png\" alt=\"3013.png\" width=\"24\" height=\"34\" \/>= 2.179<\/li>\n<\/ul>\n<\/div>\n<p>The critical values from the students\u2019 t-distribution approach the critical values from the standard normal distribution as the sample size (n) increases.<\/p>\n<div style=\"width: 508px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-57\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170716\/3002.png\" alt=\"3002.png\" width=\"498\" height=\"201\" \/><\/p>\n<p class=\"wp-caption-text\">Table 3. Critical values from the student\u2019s t-table.<\/p>\n<\/div>\n<p>Using the standard normal curve, the critical value for a 95% confidence interval is <strong class=\"Strong-2\">1.96<\/strong>. You can see how different samples sizes will change the critical value and thus the confidence interval, especially when the sample size is small.<\/p>\n<h3>Construction of a Confidence Interval about <strong class=\"char-style-override-2\">\u03bc<\/strong> when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Unknown<\/h3>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170718\/2995.png\" alt=\"2995.png\" width=\"28\" height=\"39\" \/><\/span> critical value with n-1 df<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170719\/2987.png\" alt=\"2987.png\" width=\"103\" height=\"53\" \/><\/span><\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170720\/2979.png\" alt=\"2979.png\" width=\"50\" height=\"24\" \/><\/span><\/li>\n<\/ol>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p class=\"Example\">Researchers studying the effects of acid rain in the Adirondack Mountains collected water samples from 22 lakes. They measured the pH (acidity) of the water and want to construct a 99% confidence interval about the mean lake pH for this region. The sample mean is 6.4438 with a sample standard deviation of 0.7120. They do not know anything about the distribution of the pH of this population, and the sample is small (n&lt;30), so they look at a normal probability plot.<\/p>\n<div style=\"width: 911px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-13\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170721\/2970.png\" alt=\"2970.png\" width=\"901\" height=\"600\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. Normal probability plot.<\/p>\n<\/div>\n<p class=\"Example\">The data is normally distributed. Now construct the 99% confidence interval about the mean pH.<\/p>\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170723\/2963.png\" alt=\"2963.png\" width=\"29\" height=\"41\" \/><\/span> = 2.831<\/p>\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170724\/2955.png\" alt=\"2955.png\" width=\"115\" height=\"59\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170725\/2948.png\" alt=\"2948.png\" width=\"120\" height=\"54\" \/><\/span>= 0.4297<\/p>\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170726\/2940.png\" alt=\"2940.png\" width=\"53\" height=\"25\" \/><\/span> = 6.443 \u00b1 0.4297<\/p>\n<p class=\"Example\">The 99% confidence interval about the mean pH is (6.013, 6.863).<\/p>\n<p class=\"Example\">We are 99% confident that this interval contains the mean lake pH for this lake population.<\/p>\n<p class=\"Example\">Now construct a 90% confidence interval about the mean pH for these lakes.<\/p>\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170726\/2929.png\" alt=\"2929.png\" width=\"30\" height=\"42\" \/><\/span> = 1.721<\/p>\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170727\/2919.png\" alt=\"2919.png\" width=\"117\" height=\"60\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170728\/2909.png\" alt=\"2909.png\" width=\"119\" height=\"55\" \/><\/span>= 0.2612<\/p>\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170729\/2900.png\" alt=\"2900.png\" width=\"55\" height=\"26\" \/><\/span> = 6.443 \u00b1 0.2612<\/p>\n<p class=\"Example\">The 90% confidence interval about the mean pH is (6.182, 6.704).<\/p>\n<p class=\"Example\">We are 90% confident that this interval contains the mean lake pH for this lake population.<\/p>\n<p class=\"Example\">Notice how the width of the interval decreased as the level of confidence decreased from 99 to 90%.<\/p>\n<p class=\"Example\">Construct a 90% confidence interval about the mean lake pH using Excel and Minitab.<\/p>\n<\/div>\n<h2>Software Solutions<\/h2>\n<h3>Minitab<\/h3>\n<p class=\"Example\">For Minitab, enter the data in the spreadsheet and select Basic statistics and 1-sample t-test.<\/p>\n<p class=\"No-Caption\"><img decoding=\"async\" class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170731\/035_2_fmt.png\" alt=\"035_2.tif\" \/><img decoding=\"async\" class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170734\/035_1_fmt.png\" alt=\"035_1.tif\" \/><\/p>\n<h4>One-Sample T: pH<\/h4>\n<table class=\"Table\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 43px\">\n<td class=\"Table\" style=\"height: 43px\">Variable<\/td>\n<td class=\"Table\" style=\"height: 43px\">N<\/td>\n<td class=\"Table\" style=\"height: 43px\">Mean<\/td>\n<td class=\"Table\" style=\"height: 43px\">StDev<\/td>\n<td class=\"Table\" style=\"height: 43px\">SE Mean<\/td>\n<td class=\"Table\" style=\"height: 43px\">90% CI<\/td>\n<\/tr>\n<tr style=\"height: 43.1875px\">\n<td class=\"Table\" style=\"height: 43.1875px\">pH<\/td>\n<td class=\"Table\" style=\"height: 43.1875px\">\n<p class=\"Table\">22<\/p>\n<\/td>\n<td class=\"Table\" style=\"height: 43.1875px\">\n<p class=\"Table\">6.443<\/p>\n<\/td>\n<td class=\"Table\" style=\"height: 43.1875px\">\n<p class=\"Table\">0.712<\/p>\n<\/td>\n<td class=\"Table\" style=\"height: 43.1875px\">\n<p class=\"Table\">0.152<\/p>\n<\/td>\n<td class=\"Table\" style=\"height: 43.1875px\">\n<p class=\"Table\">(6.182, 6.704)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"Example\">Additional example:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Statistics - Constructing a Confidence Model\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/gIyPlEJE6Jc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>For Excel, enter the data in the spreadsheet and select descriptive statistics. Check Summary Statistics and select the level and confidence.Excel<\/h3>\n<p class=\"No-Caption\"><span class=\"Equation-Right\"><img decoding=\"async\" class=\"frame-52\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170737\/034_2_fmt.png\" alt=\"034_2.tif\" \/><\/span><span class=\"Equation-Left\"><img decoding=\"async\" class=\"frame-29\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170740\/034_1_fmt.png\" alt=\"034_1.tif\" \/><\/span><\/p>\n<table id=\"table-3\" class=\"Table\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>\n<p class=\"Table\">Mean<\/p>\n<\/td>\n<td>\n<p class=\"Table\">6.442909<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Standard Error<\/p>\n<\/td>\n<td>\n<p class=\"Table\">0.151801<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Median<\/p>\n<\/td>\n<td>\n<p class=\"Table\">6.4925<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Mode<\/p>\n<\/td>\n<td>\n<p class=\"Table\">#N\/A<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Standard Deviation<\/p>\n<\/td>\n<td>\n<p class=\"Table\">0.712008<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Sample Variance<\/p>\n<\/td>\n<td>\n<p class=\"Table\">0.506956<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Kurtosis<\/p>\n<\/td>\n<td>\n<p class=\"Table\">-0.5007<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Skewness<\/p>\n<\/td>\n<td>\n<p class=\"Table\">-0.60591<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Range<\/p>\n<\/td>\n<td>\n<p class=\"Table\">2.338<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Minimum<\/p>\n<\/td>\n<td>\n<p class=\"Table\">5.113<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Maximum<\/p>\n<\/td>\n<td>\n<p class=\"Table\">7.451<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Sum<\/p>\n<\/td>\n<td>\n<p class=\"Table\">141.744<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p class=\"Table\">Count<\/p>\n<\/td>\n<td>\n<p class=\"Table\">22<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"Table\">\n<p class=\"Table\">Confidence Level(90.0%)<\/p>\n<\/td>\n<td class=\"Table\">\n<p class=\"Table\">0.26121<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"Example\">Excel gives you the sample mean in the first line (6.442909) and the margin of error in the last line (0.26121). You must complete the computation yourself to obtain the interval (6.442909\u00b10.26121).<\/p>\n<h3>Confidence Intervals about the Population Proportion (<em class=\"char-style-override-2\">p<\/em>)<\/h3>\n<p>Frequently, we are interested in estimating the population proportion (<span class=\"BoldItalic Strong-2\">p<\/span>), instead of the population mean (<strong class=\"Strong-2\">\u00b5<\/strong>). For example, you may need to estimate the proportion of trees infected with beech bark disease, or the proportion of people who support \u201cgreen\u201d products. The parameter <span class=\"BoldItalic Strong-2\">p<\/span> can be estimated in the same ways as we estimated <strong class=\"Strong-2\">\u00b5<\/strong>, the population mean.<\/p>\n<h4>The Sample Proportion<\/h4>\n<ul>\n<li class=\"List-Paragraph\">The sample proportion is the best point estimate for the true population proportion.<\/li>\n<li class=\"List-Paragraph\">Sample proportion\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"frame-6\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170741\/2862.png\" alt=\"2862.png\" width=\"40\" height=\"41\" \/>\u00a0where <em>x<\/em> is the number of elements in the sample with the characteristic you are interested in, and <em>n<\/em> is the sample size.<\/li>\n<\/ul>\n<h4>The Assumption of Normality when Estimating Proportions<\/h4>\n<ul>\n<li class=\"List-Paragraph\">The assumption of a normally distributed population is still important, even though the parameter has changed.<\/li>\n<li class=\"List-Paragraph\">Normality can be verified if:<\/li>\n<li>\n<p class=\"Centered\"><span class=\"Picture\"><img decoding=\"async\" class=\"frame-58 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170742\/2856.png\" alt=\"2856.png\" \/><\/span><\/p>\n<\/li>\n<\/ul>\n<h4>Constructing a Confidence Interval about the Population Proportion<\/h4>\n<p>Constructing a confidence interval about the proportion follows the same three steps we have used in previous examples.<\/p>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170743\/2848.png\" alt=\"2848.png\" width=\"32\" height=\"35\" \/>(critical value from the standard normal table)<\/li>\n<li class=\"List-Paragraph-Number-1\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-26\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170744\/2842.png\" alt=\"2842.png\" width=\"164\" height=\"59\" \/>(margin of error)<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-18\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170744\/2836.png\" alt=\"2836.png\" width=\"54\" height=\"28\" \/><\/span> (point estimate \u00b1 margin of error)<\/li>\n<\/ol>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p class=\"ExampleHeading\">A botanist has produced a new variety of hybrid soybean that is better able to withstand drought. She wants to construct a 95% confidence interval about the germination rate (percent germination). She randomly selected 500 seeds and found that 421 have germinated.<\/p>\n<p class=\"Example\">First, compute the point estimate<\/p>\n<p class=\"ExampleCenter\"><span class=\"Picture\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-7 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170745\/2827.png\" alt=\"2827.png\" width=\"190\" height=\"58\" \/><\/span><\/p>\n<p class=\"Example\">Check normality:\u00a0<span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-53\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170746\/2818.png\" alt=\"2818.png\" width=\"415\" height=\"26\" \/><\/span><\/p>\n<p class=\"Example\">You can assume a normal distribution.<\/p>\n<p class=\"Example\">Now construct the confidence interval:<\/p>\n<p class=\"Example\">1) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-20\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170748\/2811.png\" alt=\"2811.png\" width=\"28\" height=\"30\" \/><\/span> = 1.96<\/p>\n<p class=\"Example\">2) <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-54\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170748\/2802.png\" alt=\"2802.png\" width=\"174\" height=\"55\" \/><\/span> = <span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-55\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170749\/2794.png\" alt=\"2794.png\" width=\"227\" height=\"57\" \/><\/span><\/p>\n<p class=\"Example\">3) <span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-56\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170750\/2784.png\" alt=\"2784.png\" width=\"163\" height=\"21\" \/><\/span><\/p>\n<p class=\"Example\">The 95% confidence interval for the germination rate is (81.0%, 87.4%).<\/p>\n<p class=\"Example\">We can be 95% confident that this interval contains the true germination rate for this population.<\/p>\n<\/div>\n<h2>Software Solutions<\/h2>\n<h3>Minitab<\/h3>\n<p class=\"Example\">You can use Minitab to compute the confidence interval. Select STAT&gt;Basic stats&gt;1-proportion. Select summarized data and enter the number of events (421) and the number of trials (500). Click Options and select the correct confidence level. Check \u201ctest and interval based on normal distribution\u201d if the assumption of normality has been verified.<\/p>\n<p class=\"Centered\"><img decoding=\"async\" class=\"frame-57\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170752\/036_1_fmt.png\" alt=\"036_1.tif\" \/>\u00a0<img decoding=\"async\" class=\"frame-1\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170755\/036_2_fmt.png\" alt=\"036_2.tif\" \/><\/p>\n<h4>Test and CI for One Proportion<\/h4>\n<table class=\"Table\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 43.6562px\">\n<td class=\"Table\" style=\"height: 43.6562px\">Sample<\/td>\n<td class=\"Table\" style=\"height: 43.6562px\">X<\/td>\n<td class=\"Table\" style=\"height: 43.6562px\">N<\/td>\n<td class=\"Table\" style=\"height: 43.6562px\">Sample p<\/td>\n<td class=\"Table\" style=\"height: 43.6562px\">95% CI<\/td>\n<\/tr>\n<tr style=\"height: 43px\">\n<td class=\"Table\" style=\"height: 43px\">1<\/td>\n<td class=\"Table\" style=\"height: 43px\">421<\/td>\n<td class=\"Table\" style=\"height: 43px\">500<\/td>\n<td class=\"Table\" style=\"height: 43px\">0.842000<\/td>\n<td class=\"Table\" style=\"height: 43px\">(0.810030, 0.873970)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"Example\">Using the normal approximation.<\/p>\n<h3>Excel<\/h3>\n<p class=\"Example\">Excel does not compute confidence intervals for estimating the population proportion.<\/p>\n<h3>Confidence Interval Summary<\/h3>\n<p>Which method do I use?<\/p>\n<p>The first question to ask yourself is: <strong class=\"Strong-2\">Which parameter are you trying to estimate<\/strong><strong>?<\/strong> If it is the mean (<em>\u00b5<\/em>), then ask yourself: <strong class=\"Strong-2\">Is the population standard deviation (<\/strong><strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong><strong class=\"Strong-2\">) known<\/strong><strong>?<\/strong> If yes, then follow the next 3 steps:<\/p>\n<h4>Confidence Interval about the Population Mean (\u00b5) when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Known<\/h4>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170757\/2755.png\" alt=\"2755.png\" width=\"35\" height=\"38\" \/><\/span> critical value (from the standard normal table)<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-46\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170757\/2747.png\" alt=\"2747.png\" width=\"115\" height=\"56\" \/><\/span><\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170758\/2739.png\" alt=\"2739.png\" width=\"51\" height=\"24\" \/><\/span><\/li>\n<\/ol>\n<p>If no, follow these 3 steps:<\/p>\n<h4>Confidence Interval about the Population Mean (\u00b5) when <strong class=\"SymbolsBold\" xml:lang=\"ar-SA\">\u03c3<\/strong> is Unknown<\/h4>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170759\/2731.png\" alt=\"2731.png\" width=\"28\" height=\"40\" \/> critical value with n-1 df from the student t-distribution<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-14\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170759\/2723.png\" alt=\"2723.png\" width=\"115\" height=\"59\" \/><\/span><\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170800\/2714.png\" alt=\"2714.png\" width=\"53\" height=\"25\" \/><\/span><\/li>\n<\/ol>\n<p>If you want to construct a confidence interval about the population proportion, follow these 3 steps:<\/p>\n<h4>Confidence Interval about the Proportion<\/h4>\n<ol>\n<li class=\"List-Paragraph-Number-1\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-23\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170801\/2707.png\" alt=\"2707.png\" width=\"38\" height=\"41\" \/> critical value from the standard normal table<\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation-Large\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-58\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170801\/2696.png\" alt=\"2696.png\" width=\"173\" height=\"58\" \/><\/span><\/li>\n<li class=\"List-Paragraph-Number-1\"><span class=\"Inline-Equation\"><img loading=\"lazy\" decoding=\"async\" class=\"frame-43\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1888\/2017\/05\/11170802\/2685.png\" alt=\"2685.png\" width=\"52\" height=\"25\" \/><\/span><\/li>\n<\/ol>\n<p>Remember that the assumption of normality must be verified.<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-158\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Natural Resources Biometrics. <strong>Authored by<\/strong>: Diane Kiernan. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/textbooks.opensuny.org\/natural-resources-biometrics\/\">https:\/\/textbooks.opensuny.org\/natural-resources-biometrics\/<\/a>. <strong>Project<\/strong>: Open SUNY Textbooks. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":622,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Natural Resources Biometrics\",\"author\":\"Diane Kiernan\",\"organization\":\"\",\"url\":\"https:\/\/textbooks.opensuny.org\/natural-resources-biometrics\/\",\"project\":\"Open SUNY Textbooks\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-158","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapters\/158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/wp\/v2\/users\/622"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapters\/158\/revisions"}],"predecessor-version":[{"id":1247,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapters\/158\/revisions\/1247"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapters\/158\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/wp\/v2\/media?parent=158"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/pressbooks\/v2\/chapter-type?post=158"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/wp\/v2\/contributor?post=158"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-natural-resources-biometrics\/wp-json\/wp\/v2\/license?post=158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}