{"id":5237,"date":"2022-08-19T00:16:50","date_gmt":"2022-08-19T00:16:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5237"},"modified":"2022-08-19T00:22:18","modified_gmt":"2022-08-19T00:22:18","slug":"10a-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/10a-preview\/","title":{"raw":"10A Preview","rendered":"10A Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to calculate the components of a confidence\u00a0 interval for a population proportion, including the point estimate, standard error, z critical\u00a0 values, and margin of error. You will also need to use these components to construct\u00a0 confidence intervals for proportions and represent them on number lines. To do this,\u00a0 you will need to understand the connection between sampling distributions, the Central\u00a0 Limit Theorem, and confidence intervals.\r\n
\r\n

Question 1<\/h3>\r\nSuppose we want to estimate the proportion of college students who have a high\u00a0 level of stress. In a previous homework assignment, we looked at a sample of 253\u00a0 college students and found that 56 of them had high levels of stress.[footnote]Onyper, S., Thacher, P., Gilbert, J., & Gradess, S. (2012). Class start times, sleep, and academic performance in college: A path analysis. Chronobiology International<\/em>, 29(3): 318\u2013335.[\/footnote]\r\n\r\nWhat proportion of students in the sample had a high level of stress?\r\n\r\nHint: To calculate the proportion, take the number of successes divided by the\u00a0 sample size.\r\n\r\n<\/div>\r\nRecall that in the previous activities, we assumed we knew the true population\u00a0 parameter, [latex]p[\/latex], and looked at how close sample proportions were to this value. Often, we\u00a0 will not know the true population parameter, so we will need to estimate it. The sample\u00a0 proportion is used as a point estimate of the population proportion. A point estimate is\u00a0 a single value based on representative sample data that is a plausible estimate of the\u00a0 population parameter. In this case, we are talking about proportions, so the best\u00a0 estimate for the population proportion is the sample proportion. We refer to the point\u00a0 estimate for proportions as [latex]\\hat{p}[\/latex].\r\n
\r\n

Question 2<\/h3>\r\nRefer to Question 1 to answer the following questions.\r\n
    \r\n \t
  1. What do we want to estimate?\r\n
      \r\n \t
    1. The population proportion, [latex]p [\/latex]<\/li>\r\n \t
    2. The sample proportion, [latex]\\hat{p}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
    3. What are we going to use to estimate the proportion of college students who\u00a0 have a high level of stress?\r\n
        \r\n \t
      1. The population proportion, [latex]p [\/latex]<\/li>\r\n \t
      2. The sample proportion, [latex]\\hat{p}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Question 3<\/h3>\r\nSuppose we took another different sample of 253 college students. Would you\u00a0 expect the proportion of students with a high level of stress to be the same as in\u00a0 Question 1 or different from Question 1?\r\n
          \r\n \t
        1. The same<\/li>\r\n \t
        2. Different<\/li>\r\n<\/ol>\r\n<\/div>\r\nRecall from In-Class Activity 9.C:\r\n
          \r\n\r\nSampling Distribution of the Sample Proportion\r\n\r\nWhen taking many, many random samples of size n from a population distribution with proportion [latex]p[\/latex]:\r\n\r\nThe mean of the distribution of sample proportions is [latex]p[\/latex].\r\n\r\nThe standard deviation of the distribution of sample proportions is[latex] \\sqrt{\\frac{p(1-p)}{n}} [\/latex].\r\n\r\nIf [latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the Central Limit Theorem (CLT) states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[\/latex] and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].\r\n\r\n<\/div>\r\n \r\n
          \r\n

          Question 4<\/h3>\r\nIs the sample size in Question 1 large enough to use the approximate distribution \u00a0stated by the CLT?\r\n\r\nHint: Use the condition [latex]n\\hat{p} \\geq 10[\/latex] and [latex]n(1\u2212\\hat{p}) \\geq 10[\/latex].\r\n\r\n<\/div>\r\n
          \r\n

          Question 5<\/h3>\r\nWhat can we say about how close the sample proportion is to the population proportion?\r\n
            \r\n \t
          1. Approximately 68% of the sample proportions will fall within 2 standard deviations of the mean ([latex]p[\/latex]).<\/li>\r\n \t
          2. Approximately 95% of the sample proportions will fall within 2 standard deviations of the mean ([latex]p[\/latex]).<\/li>\r\n \t
          3. Approximately 99.7% of the sample proportions will fall within 2 standard deviations of the mean ([latex]p[\/latex]).<\/li>\r\n<\/ol>\r\nHint: Remember the Empirical Rule.\r\n\r\n<\/div>\r\nRecall that when the sample size is large enough, we can use [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}} [\/latex] in place of[latex] \\sqrt{\\frac{p(1-p)}{n}} [\/latex] .\u00a0 This is called the standard error, which is the estimated standard deviation of sample proportions. It is the measure of sample-to-sample variability. We will use the standard error to help us convey information about the accuracy of our point estimate!\r\n
            \r\n

            Question 6<\/h3>\r\nCalculate the estimated standard deviation for the sample proportion for the scenario in Question 1. Round your answer to 2 decimal places.\r\n\r\n<\/div>\r\nA confidence interval for a population proportion is a reasonable range of values where we expect the population proportion to fall within, with a chosen degree of confidence. We create confidence intervals because, as seen in previous in-class activities, sample proportions vary from sample to sample.\r\n\r\nA sample proportion will most likely not be equal to the true population parameter. Thus, instead of simply using a single value (a point estimate) to estimate our parameter, we will use a range of values. Confidence intervals take sampling variability into account and convey information on estimate accuracy.\r\n\r\nA confidence interval is calculated using the point estimate and the margin of error. The margin of error (E) is what determines the width of the interval. A confidence interval will have a width of twice the margin of error. The following is a visualization of a confidence interval.\r\n\r\n\"\"\r\n\r\nThe margin of error is calculated using the standard error and the z critical value for the confidence level. This means that the width of our interval is determined by how much our sample data varies and how confident we want to be in our method.\r\n\r\nThe confidence level, [latex]C[\/latex], tells us how much confidence we have in the method used to construct the interval. It corresponds to the percentage of all intervals we would expect to contain the true population parameter.\r\n\r\nFor example, if we have a 95% confidence level and we took a very large number of different samples and created confidence intervals for each one, we would expect about 95% of them to contain the population parameter.\r\n\r\nFor proportions, each confidence level has a corresponding z critical value ([latex]z^{*}[\/latex]). In Question 5, we applied the Empirical Rule, which indicates [latex]z^{*}= 2 [\/latex]and its negative counterpart, -2. As you can see below, the values of 2 separate the middle 95.45% from the most extreme 4.55% areas. Note that the most extreme area is split evenly in each tail.\r\n\r\n\"\"\r\n\r\nIn practice, we are usually interested in 90%, 95%, and 99% confidence intervals. Below is the standard normal distribution displaying the z critical value, [latex]z^{*}[\/latex], for a 95% confidence level. As you can see, 95% of the curve is shaded in and the remaining 5% of the curve is unshaded in the tails. The values that separate the middle 95% from the most extreme 5% are what we will use for our confidence interval.\r\n
            [latex]z^{*}[\/latex]: The z critical value; this is the point on the standard normal distribution such that the proportion of area under the curve between [latex]-z^{*}[\/latex] and [latex]+z^{*}[\/latex] is [latex]C [\/latex], the confidence level.<\/div>\r\nThe\u00a0[latex]z^{*}[\/latex] value is found using the DCMP Normal Distribution tool at https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/<\/a>.\r\n