{"id":507,"date":"2017-04-15T03:26:03","date_gmt":"2017-04-15T03:26:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/estimating-a-population-mean-3-of-3\/"},"modified":"2017-05-31T03:44:07","modified_gmt":"2017-05-31T03:44:07","slug":"estimating-a-population-mean-3-of-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/chapter\/estimating-a-population-mean-3-of-3\/","title":{"raw":"Estimating a Population Mean (3 of 3)","rendered":"Estimating a Population Mean (3 of 3)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Construct a confidence interval to estimate a population mean when conditions are met. Interpret the confidence interval in context.<\/li>\r\n \t<li>Adjust the margin of error by making changes to the confidence level or sample size.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Structure of a Confidence Interval<\/h3>\r\nLet\u2019s take a closer look at the parts of the confidence interval. Remember that this is a confidence interval for a population mean. We use this formula when the population standard deviation is unknown.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032548\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image1.png\" alt=\"Sample mean and center of interval = Critical T-value * Standard Error\" width=\"329\" height=\"195\" \/>\r\n\r\nLet\u2019s remind ourselves how the confidence interval formula relates to the graph of the confidence interval on a number line.\r\n\r\nThe confidence interval shown below is a 95% confidence interval for a sample of size <em>n<\/em> = 25 (so <em>df<\/em> = 24), with sample mean [latex]\\overline{x}[\/latex] = 9 and sample standard deviation of <em>s<\/em> = 3. The critical T-value for a 95% confidence interval with a <em>df<\/em> = 24 is 2.064.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\mathrm{Standard}\\text{}\\mathrm{error}\\text{}\\mathrm{is}\\text{}3\\text{}&amp;sol;\\sqrt{25}\\text{}=\\text{}0.6\\\\ \\mathrm{Margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}\\text{}(\\mathrm{ME})\\text{}\\mathrm{is}\\text{}2.064(3)\\text{}&amp;sol;\\text{}\\sqrt{25}\\text{}\\approx \\text{}1.24\\end{array}[\/latex]<\/p>\r\nThe confidence interval is 9 \u00b1 1.24. We are 95% confident that \u00b5 lies between 7.76 and 10.24.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032551\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image3.png\" alt=\"A number line. Highlighted is the sample mean - ME and the sample mean + ME, with the sample mean of 9 marked. The width of the interval is 2 * ME.\" width=\"559\" height=\"304\" \/>\r\n\r\nNote:\r\n<ul>\r\n \t<li>The sample mean (9 in this example) is at the center of the interval.<\/li>\r\n \t<li>The margin of error (labeled ME and equal to 1.24 in this example) is the distance that the interval extends to the left and right of the sample mean.<\/li>\r\n \t<li>The interval width is the length of the entire interval on the number line. The interval width is always twice the margin of error.<\/li>\r\n<\/ul>\r\nLet\u2019s quickly review how the <em>precision <\/em>of a confidence interval relates to the margin of error:\r\n<ul>\r\n \t<li>An interval gives a <em>more precise <\/em>estimate when the interval is narrower. In other words, the margin of error is smaller.<\/li>\r\n \t<li>An interval gives a <em>less precise <\/em>estimate when the interval is wider. In other words, the margin of error is larger.<\/li>\r\n<\/ul>\r\nWe know that a higher confidence level gives a larger margin of error, so confidence level is also related to precision.\r\n<ul>\r\n \t<li>Increasing the confidence in our estimate makes the confidence interval wider and therefore less precise.<\/li>\r\n \t<li>Decreasing the confidence in our estimate makes the confidence interval narrower, and therefore more precise.<\/li>\r\n<\/ul>\r\nConfidence interval estimates are useful when they have the right balance of confidence and precision. Typical confidence levels used in practice are 90%, 95%, and 99%. When we need to be really sure about our estimates, such as in life-and-death situations, we choose a 99% confidence level. So if nothing else changes, we settle for less precise estimates when we need a high level of confidence.\r\n<h3><\/h3>\r\nIn our discussion about the structure of confidence intervals, we said choosing a higher level of confidence means that we sacrifice some precision. This is true only if nothing else changes. But there is one way to keep a high level of confidence without sacrificing precision: Increase the sample size. We investigate the impact of sample size on the confidence interval next.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>Cable Strength Revisited<\/h2>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032556\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_cable.jpg\" alt=\"Cable\" width=\"279\" height=\"430\" \/>\r\n\r\nRecall the engineers who are trying to determine the breaking weight of a cable. In that example, we had a random sample of 45 cables with a mean breaking weight of 768.2 lb and a standard deviation of 15.1 lb. From that sample we computed a 95% confidence interval for the mean breaking weight of all such cables. Here are the important numbers we found from that calculation on the previous page:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\mathrm{standard}\\text{}\\mathrm{error}:\\text{}s\\text{}&amp;sol;\\sqrt{n}\\text{}=\\text{}15.1\\text{}&amp;sol;\\sqrt{45}\\text{}\\approx \\text{}2.25\\\\ \\mathrm{critical}\\text{}\\text{T-value}:\\text{}{T}_{c}\\text{}=\\text{}2.015\\text{}(\\mathrm{we}\\text{}\\mathrm{found}\\text{}\\mathrm{this}\\text{}\\mathrm{using}\\text{}\\mathrm{the}\\text{}\\mathrm{simulation})\\\\ \\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}:\\text{}{T}_{c}\\text{}\u22c5\\text{}s\\text{}&amp;sol;\\sqrt{n}\\text{}=\\text{}2.015(2.25)\\text{}=\\text{}4.53\\\\ \\mathrm{confidence}\\text{}\\mathrm{interval}:\\text{}768.2\\text{}&amp;PlusMinus;\\text{}4.53\\text{}\\mathrm{or}\\text{}(763.67,772.73)\\end{array}[\/latex]<\/p>\r\nNow let\u2019s increase the sample size and investigate the impact on the confidence interval. We calculate the confidence interval for a larger sample of 101 cables (<em>n<\/em> = 101).\r\n\r\nSample size affects our calculations in two ways:\r\n<ul>\r\n \t<li>The sample size (<em>n<\/em>) appears in our formula for standard error.<\/li>\r\n \t<li>The critical T-value depends on degrees of freedom, and <em>df<\/em> = <em>n<\/em> - 1.<\/li>\r\n<\/ul>\r\n<strong>Finding the standard error:<\/strong>\r\n\r\nWe approximate the standard error of all sample means as follows:\r\n<p style=\"text-align: center\">[latex]s\\text{}&amp;sol;\\sqrt{n}\\text{}=\\text{}15.1\\text{}&amp;sol;\\text{}\\sqrt{101}\\text{}\\approx \\text{}1.50[\/latex]<\/p>\r\nNote: The standard error is smaller when the sample size is larger. We were expecting this because we know there is less variability in sample means when the samples are larger.\r\n\r\n<strong>Finding the critical T-value:<\/strong>\r\n\r\nTo find the critical T-value, we use the simulation. We set the <em>df<\/em> to 100 and the central probability to 0.95. We see that the critical T-value is 1.984.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032559\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image12.png\" alt=\"A bell curve centered at 0 with the center 95% shaded in green underneath it.\" width=\"398\" height=\"306\" \/>\r\n\r\nNote: Increasing the sample size decreased the critical T-value (the T-value went from 2.015 to 1.984 when we increased the sample size). You might also notice that both of the critical T-values for 95% confidence are larger than the critical Z-value for 95% confidence, which is approximately 1.96. This makes sense because the T-models are wider than the the standard normal curve.\r\n\r\n<strong>Finding the margin of error.<\/strong>\r\n\r\nHere is the margin of error calculation:\r\n<p style=\"text-align: center\">[latex]{T}_{c}\\text{}\u22c5\\text{}s\\text{}&amp;sol;\\text{}\\sqrt{n}=1.984(1.50)\\text{}=\\text{}2.98[\/latex]<\/p>\r\n<strong>Finding the confidence interval.<\/strong>\r\n\r\nHere is the confidence interval calculation:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\overline{x}\\text{}&amp;PlusMinus;\\text{}\\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}\\\\ \\overline{x}\\text{}&amp;PlusMinus;\\text{}{T}_{c}\\text{}\u22c5\\text{}\\frac{s}{\\sqrt{n}}\\\\ 768.2\\text{}&amp;PlusMinus;\\text{}2.98\\\\ (765.22,771.18)\\end{array}[\/latex]<\/p>\r\n<strong>Side-by-side comparison:<\/strong>\r\n\r\nLet\u2019s take a look at these two intervals to study the effects of changing the sample size.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032601\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image15.png\" alt=\"Table showing: sample size, standard error, critical t-value, margin of error, and confidence interval. For a sample size where n=45: Standard error is 2.25, critical t-value is 2.015, margin of error is 4.53, and confidence interval is (763.67, 772.73). For a sample size where n=45: Standard error is 1.50, critical t-value is 1.984, margin of error is 2.98, and confidence interval is (765.22, 771.18).\" width=\"447\" height=\"159\" \/>\r\n\r\nIncreasing the sample size had the following effects on the confidence interval estimate:\r\n<ul>\r\n \t<li>Decreased standard error<\/li>\r\n \t<li>Decreased critical T-value<\/li>\r\n \t<li>Decreased margin of error and hence decreased the interval width<\/li>\r\n \t<li>Improved interval precision<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Comment<\/h3>\r\nIn the real world, increasing the sample size is not always possible. Sometimes collecting a sample is very expensive. If the study has budgetary constraints, which is usually the case, selecting a larger sample may be too expensive.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\n<h2>Appropriate Conclusions<\/h2>\r\nFor each of the following situations, decide if it is valid or invalid to use a confidence interval to estimate the population mean.\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3688\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3689\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3690\r\n\r\n<\/div>\r\n<h3>Let\u2019s Summarize<\/h3>\r\n<ul>\r\n \t<li>A confidence interval approximates a population mean by giving us a range of values that likely contains the population mean, \u03bc. The general form of the confidence interval is [latex]\\stackrel{\u00af}{x}\\text{}&amp;PlusMinus;\\text{}\\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}.[\/latex]<\/li>\r\n \t<li>To say that we are \u201c95% confident that the population mean falls within our confidence interval\u201d really means that \u201cabout 95% of all confidence intervals computed in this way will capture the true population mean.\u201d<\/li>\r\n \t<li>We can use a sample mean to build a confidence interval as an estimate for \u03bc. There are two possible cases:\r\n<ul>\r\n \t<li>Suppose the population standard deviation, \u03c3, is known. We check the conditions for use of the normal model. Conditions: The variable must be normally distributed in the population, or the sample size is large enough (<em>n<\/em>\u00a0&amp;gt; 30). In this case, the confidence interval has the form [latex]\\stackrel{\u00af}{x}\\text{}&amp;PlusMinus;\\text{}{Z}_{c}\\text{}\u22c5\\text{}\u03c3\\text{}&amp;sol;\\text{}\\sqrt{n}[\/latex].<\/li>\r\n \t<li>Suppose the population standard deviation, \u03c3, is not known. Then we use the sample standard deviation, <em>s<\/em>, as an approximation for \u03c3. We check the conditions for use of the T-model. Conditions are the same: The variable must be normally distributed in the population, or the sample size is large enough (<em>n<\/em>\u00a0&amp;gt; 30). In this case, the confidence interval has the form [latex]\\stackrel{\u00af}{x}\\text{}&amp;PlusMinus;\\text{}{T}_{c}\\text{}\u22c5\\text{}s\\text{}&amp;sol;\\text{}\\sqrt{n}[\/latex] .When using the T-model to find the critical value, we need to select an appropriate number of degrees of freedom (<em>df<\/em>). The number of degrees of freedom is 1 less than the sample size (<em>df<\/em> = <em>n<\/em> - 1).<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>As we have seen with other confidence intervals, the width of a confidence interval is twice the margin of error. The smaller the margin of error, the more narrow the confidence interval and the more precise the estimate of \u00b5.<\/li>\r\n<\/ul>\r\n<h3><\/h3>","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Construct a confidence interval to estimate a population mean when conditions are met. Interpret the confidence interval in context.<\/li>\n<li>Adjust the margin of error by making changes to the confidence level or sample size.<\/li>\n<\/ul>\n<\/div>\n<h3>Structure of a Confidence Interval<\/h3>\n<p>Let\u2019s take a closer look at the parts of the confidence interval. Remember that this is a confidence interval for a population mean. We use this formula when the population standard deviation is unknown.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032548\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image1.png\" alt=\"Sample mean and center of interval = Critical T-value * Standard Error\" width=\"329\" height=\"195\" \/><\/p>\n<p>Let\u2019s remind ourselves how the confidence interval formula relates to the graph of the confidence interval on a number line.<\/p>\n<p>The confidence interval shown below is a 95% confidence interval for a sample of size <em>n<\/em> = 25 (so <em>df<\/em> = 24), with sample mean [latex]\\overline{x}[\/latex] = 9 and sample standard deviation of <em>s<\/em> = 3. The critical T-value for a 95% confidence interval with a <em>df<\/em> = 24 is 2.064.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\mathrm{Standard}\\text{}\\mathrm{error}\\text{}\\mathrm{is}\\text{}3\\text{}&sol;\\sqrt{25}\\text{}=\\text{}0.6\\\\ \\mathrm{Margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}\\text{}(\\mathrm{ME})\\text{}\\mathrm{is}\\text{}2.064(3)\\text{}&sol;\\text{}\\sqrt{25}\\text{}\\approx \\text{}1.24\\end{array}[\/latex]<\/p>\n<p>The confidence interval is 9 \u00b1 1.24. We are 95% confident that \u00b5 lies between 7.76 and 10.24.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032551\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image3.png\" alt=\"A number line. Highlighted is the sample mean - ME and the sample mean + ME, with the sample mean of 9 marked. The width of the interval is 2 * ME.\" width=\"559\" height=\"304\" \/><\/p>\n<p>Note:<\/p>\n<ul>\n<li>The sample mean (9 in this example) is at the center of the interval.<\/li>\n<li>The margin of error (labeled ME and equal to 1.24 in this example) is the distance that the interval extends to the left and right of the sample mean.<\/li>\n<li>The interval width is the length of the entire interval on the number line. The interval width is always twice the margin of error.<\/li>\n<\/ul>\n<p>Let\u2019s quickly review how the <em>precision <\/em>of a confidence interval relates to the margin of error:<\/p>\n<ul>\n<li>An interval gives a <em>more precise <\/em>estimate when the interval is narrower. In other words, the margin of error is smaller.<\/li>\n<li>An interval gives a <em>less precise <\/em>estimate when the interval is wider. In other words, the margin of error is larger.<\/li>\n<\/ul>\n<p>We know that a higher confidence level gives a larger margin of error, so confidence level is also related to precision.<\/p>\n<ul>\n<li>Increasing the confidence in our estimate makes the confidence interval wider and therefore less precise.<\/li>\n<li>Decreasing the confidence in our estimate makes the confidence interval narrower, and therefore more precise.<\/li>\n<\/ul>\n<p>Confidence interval estimates are useful when they have the right balance of confidence and precision. Typical confidence levels used in practice are 90%, 95%, and 99%. When we need to be really sure about our estimates, such as in life-and-death situations, we choose a 99% confidence level. So if nothing else changes, we settle for less precise estimates when we need a high level of confidence.<\/p>\n<h3><\/h3>\n<p>In our discussion about the structure of confidence intervals, we said choosing a higher level of confidence means that we sacrifice some precision. This is true only if nothing else changes. But there is one way to keep a high level of confidence without sacrificing precision: Increase the sample size. We investigate the impact of sample size on the confidence interval next.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>Cable Strength Revisited<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032556\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_cable.jpg\" alt=\"Cable\" width=\"279\" height=\"430\" \/><\/p>\n<p>Recall the engineers who are trying to determine the breaking weight of a cable. In that example, we had a random sample of 45 cables with a mean breaking weight of 768.2 lb and a standard deviation of 15.1 lb. From that sample we computed a 95% confidence interval for the mean breaking weight of all such cables. Here are the important numbers we found from that calculation on the previous page:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\mathrm{standard}\\text{}\\mathrm{error}:\\text{}s\\text{}&sol;\\sqrt{n}\\text{}=\\text{}15.1\\text{}&sol;\\sqrt{45}\\text{}\\approx \\text{}2.25\\\\ \\mathrm{critical}\\text{}\\text{T-value}:\\text{}{T}_{c}\\text{}=\\text{}2.015\\text{}(\\mathrm{we}\\text{}\\mathrm{found}\\text{}\\mathrm{this}\\text{}\\mathrm{using}\\text{}\\mathrm{the}\\text{}\\mathrm{simulation})\\\\ \\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}:\\text{}{T}_{c}\\text{}\u22c5\\text{}s\\text{}&sol;\\sqrt{n}\\text{}=\\text{}2.015(2.25)\\text{}=\\text{}4.53\\\\ \\mathrm{confidence}\\text{}\\mathrm{interval}:\\text{}768.2\\text{}&PlusMinus;\\text{}4.53\\text{}\\mathrm{or}\\text{}(763.67,772.73)\\end{array}[\/latex]<\/p>\n<p>Now let\u2019s increase the sample size and investigate the impact on the confidence interval. We calculate the confidence interval for a larger sample of 101 cables (<em>n<\/em> = 101).<\/p>\n<p>Sample size affects our calculations in two ways:<\/p>\n<ul>\n<li>The sample size (<em>n<\/em>) appears in our formula for standard error.<\/li>\n<li>The critical T-value depends on degrees of freedom, and <em>df<\/em> = <em>n<\/em> &#8211; 1.<\/li>\n<\/ul>\n<p><strong>Finding the standard error:<\/strong><\/p>\n<p>We approximate the standard error of all sample means as follows:<\/p>\n<p style=\"text-align: center\">[latex]s\\text{}&sol;\\sqrt{n}\\text{}=\\text{}15.1\\text{}&sol;\\text{}\\sqrt{101}\\text{}\\approx \\text{}1.50[\/latex]<\/p>\n<p>Note: The standard error is smaller when the sample size is larger. We were expecting this because we know there is less variability in sample means when the samples are larger.<\/p>\n<p><strong>Finding the critical T-value:<\/strong><\/p>\n<p>To find the critical T-value, we use the simulation. We set the <em>df<\/em> to 100 and the central probability to 0.95. We see that the critical T-value is 1.984.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032559\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image12.png\" alt=\"A bell curve centered at 0 with the center 95% shaded in green underneath it.\" width=\"398\" height=\"306\" \/><\/p>\n<p>Note: Increasing the sample size decreased the critical T-value (the T-value went from 2.015 to 1.984 when we increased the sample size). You might also notice that both of the critical T-values for 95% confidence are larger than the critical Z-value for 95% confidence, which is approximately 1.96. This makes sense because the T-models are wider than the the standard normal curve.<\/p>\n<p><strong>Finding the margin of error.<\/strong><\/p>\n<p>Here is the margin of error calculation:<\/p>\n<p style=\"text-align: center\">[latex]{T}_{c}\\text{}\u22c5\\text{}s\\text{}&sol;\\text{}\\sqrt{n}=1.984(1.50)\\text{}=\\text{}2.98[\/latex]<\/p>\n<p><strong>Finding the confidence interval.<\/strong><\/p>\n<p>Here is the confidence interval calculation:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\overline{x}\\text{}&PlusMinus;\\text{}\\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}\\\\ \\overline{x}\\text{}&PlusMinus;\\text{}{T}_{c}\\text{}\u22c5\\text{}\\frac{s}{\\sqrt{n}}\\\\ 768.2\\text{}&PlusMinus;\\text{}2.98\\\\ (765.22,771.18)\\end{array}[\/latex]<\/p>\n<p><strong>Side-by-side comparison:<\/strong><\/p>\n<p>Let\u2019s take a look at these two intervals to study the effects of changing the sample size.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032601\/m10_inference_mean_topic_10_2_m10_est_pop_means_3_image15.png\" alt=\"Table showing: sample size, standard error, critical t-value, margin of error, and confidence interval. For a sample size where n=45: Standard error is 2.25, critical t-value is 2.015, margin of error is 4.53, and confidence interval is (763.67, 772.73). For a sample size where n=45: Standard error is 1.50, critical t-value is 1.984, margin of error is 2.98, and confidence interval is (765.22, 771.18).\" width=\"447\" height=\"159\" \/><\/p>\n<p>Increasing the sample size had the following effects on the confidence interval estimate:<\/p>\n<ul>\n<li>Decreased standard error<\/li>\n<li>Decreased critical T-value<\/li>\n<li>Decreased margin of error and hence decreased the interval width<\/li>\n<li>Improved interval precision<\/li>\n<\/ul>\n<\/div>\n<h3>Comment<\/h3>\n<p>In the real world, increasing the sample size is not always possible. Sometimes collecting a sample is very expensive. If the study has budgetary constraints, which is usually the case, selecting a larger sample may be too expensive.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<h2>Appropriate Conclusions<\/h2>\n<p>For each of the following situations, decide if it is valid or invalid to use a confidence interval to estimate the population mean.<\/p>\n<p>\t<iframe id=\"lumen_assessment_3688\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3688&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3688\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3689\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3689&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3689\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3690\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3690&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3690\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<h3>Let\u2019s Summarize<\/h3>\n<ul>\n<li>A confidence interval approximates a population mean by giving us a range of values that likely contains the population mean, \u03bc. The general form of the confidence interval is [latex]\\stackrel{\u00af}{x}\\text{}&PlusMinus;\\text{}\\mathrm{margin}\\text{}\\mathrm{of}\\text{}\\mathrm{error}.[\/latex]<\/li>\n<li>To say that we are \u201c95% confident that the population mean falls within our confidence interval\u201d really means that \u201cabout 95% of all confidence intervals computed in this way will capture the true population mean.\u201d<\/li>\n<li>We can use a sample mean to build a confidence interval as an estimate for \u03bc. There are two possible cases:\n<ul>\n<li>Suppose the population standard deviation, \u03c3, is known. We check the conditions for use of the normal model. Conditions: The variable must be normally distributed in the population, or the sample size is large enough (<em>n<\/em>\u00a0&amp;gt; 30). In this case, the confidence interval has the form [latex]\\stackrel{\u00af}{x}\\text{}&PlusMinus;\\text{}{Z}_{c}\\text{}\u22c5\\text{}\u03c3\\text{}&sol;\\text{}\\sqrt{n}[\/latex].<\/li>\n<li>Suppose the population standard deviation, \u03c3, is not known. Then we use the sample standard deviation, <em>s<\/em>, as an approximation for \u03c3. We check the conditions for use of the T-model. Conditions are the same: The variable must be normally distributed in the population, or the sample size is large enough (<em>n<\/em>\u00a0&amp;gt; 30). In this case, the confidence interval has the form [latex]\\stackrel{\u00af}{x}\\text{}&PlusMinus;\\text{}{T}_{c}\\text{}\u22c5\\text{}s\\text{}&sol;\\text{}\\sqrt{n}[\/latex] .When using the T-model to find the critical value, we need to select an appropriate number of degrees of freedom (<em>df<\/em>). The number of degrees of freedom is 1 less than the sample size (<em>df<\/em> = <em>n<\/em> &#8211; 1).<\/li>\n<\/ul>\n<\/li>\n<li>As we have seen with other confidence intervals, the width of a confidence interval is twice the margin of error. The smaller the margin of error, the more narrow the confidence interval and the more precise the estimate of \u00b5.<\/li>\n<\/ul>\n<h3><\/h3>\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-507\">\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>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/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":163,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"ac81c3a9-a8b2-43f8-b262-4cad6c1477cb","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-507","chapter","type-chapter","status-publish","hentry"],"part":474,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/507","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/507\/revisions"}],"predecessor-version":[{"id":1490,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/507\/revisions\/1490"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/parts\/474"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/507\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/media?parent=507"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapter-type?post=507"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/contributor?post=507"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/license?post=507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}