{"id":5285,"date":"2022-08-19T15:06:00","date_gmt":"2022-08-19T15:06:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5285"},"modified":"2022-08-19T15:20:31","modified_gmt":"2022-08-19T15:20:31","slug":"10d-in-class-activity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/10d-in-class-activity\/","title":{"raw":"10D In-Class Activity","rendered":"10D In-Class Activity"},"content":{"raw":"The goal of this in-class activity is to investigate if there are differences in voting\u00a0 behaviors between eligible voters who affiliate with one of the two major political parties\u00a0 in the United States (Democrat and Republican) and those who do not.\u00a0\u00a0Specifically, we will look at the difference in the\u00a0\u00a0proportions of \u201cregular voters\u201d between the two\u00a0\u00a0groups. In these data, a \u201cregular voter\u201d is a\u00a0\u00a0person who indicated that they voted in \u201call or\u00a0\u00a0all-but-one of the elections they were eligible \u00a0for\u201d in the survey.\r\n\r\nThe authors in the FiveThirtyEight article \u201cWhy\u00a0\u00a0Many Americans Don\u2019t Vote\u201d[footnote]Thomson-DeVeaux, A., Mithani, J., & Bronner, L. (2020, October 26). Why many Americans don\u2019t vote<\/em>. FiveThirtyEight. https:\/\/projects.fivethirtyeight.com\/non-voters-poll-2020-election\/[\/footnote] limited their \u00a0analysis to only include survey responses from \u00a0voters who were eligible for at least four election cycles. For this activity, we will also limit \u00a0our data to this group.\r\n\r\n\"\"\r\n\r\nCredit: iStock\/LifeSyleVisuals\r\n
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Question 1<\/h3>\r\nWhy do you think the authors chose to only include survey responses from people \u00a0who were eligible to vote for at least four election cycles in their analysis?\r\n\r\n<\/div>\r\nThe data come from the data journalism website FiveThirtyEight. The data were originally collected as part of an online survey conducted by Ipsos, where respondents answered demographic questions along with questions about political party affiliation and voting behavior.\r\n\r\nThe data contain information for 3,594 respondents who said they identify with a major party (Republican or Democrat) and 2,242 respondents who said they do not have a major party affiliation. All the respondents had been eligible to vote for at least four election cycles at the time of the survey. The primary variables of interest are:\r\n\r\nparty_id: Republican, Democrat, or Other\r\n\r\nmajor_party: Yes if the respondent identified as Republican or Democrat; no otherwise\r\n\r\nregular_voter: Yes if the respondent voted in all or all-but-one of the elections they were eligible for; no otherwise\r\n\r\nThe first 15 observations from the dataset are presented in the table below.\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
party_id<\/td>\r\nmajor_party<\/td>\r\nregular_voter<\/td>\r\n<\/tr>\r\n
Democrat<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Other<\/td>\r\nNo<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Democrat<\/td>\r\nYes<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Democrat<\/td>\r\nYes<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Other<\/td>\r\nNo<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Democrat<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Other<\/td>\r\nNo<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Democrat<\/td>\r\nYes<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Other<\/td>\r\nNo<\/td>\r\nNo<\/td>\r\n<\/tr>\r\n
Republican<\/td>\r\nYes<\/td>\r\nYes<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOf the 3,594 respondents with a major party affiliation,1,255 were regular voters. Of the 2,242 respondents with no major party affiliation, 556 were regular voters.\r\n\r\n \r\n\r\n<\/div>\r\n
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Question 2<\/h3>\r\nOur primary goal is to use the data to examine whether there\u2019s a difference in the\u00a0 proportions of regular voters between eligible voters who said they affiliate with a\u00a0 major political party and those who said they don\u2019t.\r\n\r\nLet\u2019s start by looking at a graphical display of the two variables. Select the \u201cVoter\u201d\u00a0 dataset in the DCMP Compare Two Population Proportions tool at\r\n\r\nhttps:\/\/dcmathpathways.shinyapps.io\/2sample_prop\/.\u00a0<\/a>\r\n
    \r\n \t
  1. What are the two groups in this analysis? Are they independent or\u00a0 dependent? Explain.<\/li>\r\n \t
  2. Use software to visualize the distribution of regular voters based on whether\u00a0 the respondents said they have major party affiliations. You can use the DCMP Compare Two Population Proportions tool at\r\nhttps:\/\/dcmathpathways.shinyapps.io\/2sample_prop\/.\u00a0<\/a><\/li>\r\n \t
  3. Interpret the plot. Does there appear to be a difference in the proportions of\u00a0 regular voters between eligible voters who said they have a major party\u00a0 affiliation and those who said they don\u2019t? Explain.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIt looks like the proportion of regular voters is higher for those who said they have an\u00a0 affiliation with a major party; however, the difference does not appear to be very large.\r\n
    \r\n

    Question 3<\/h3>\r\nUltimately, we want to understand the value [latex] p_{1} - p_{2}[\/latex], where\u00a0[latex] p_{1} [\/latex] is the true population proportion of eligible voters with a major party affiliation who are regular voters and\u00a0\u00a0[latex] p_{2} [\/latex] is the true population proportion of eligible voters with no major party affiliation\u00a0 who are regular voters.\r\n\r\nIdeally, we would have complete data about the two populations of interest (eligible\u00a0 voters with and without a major party affiliation) so we could calculate\u00a0[latex] p_{1} - p_{2} [\/latex] directly. However, we don\u2019t have complete data on each population, so we\u2019ll use our\u00a0 samples to draw conclusions about the differences between these two groups.\r\n
      \r\n \t
    1. What are the samples for this analysis? Include a description of each sample\u00a0 along with the sample size.<\/li>\r\n \t
    2. The sample statistic\u00a0[latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex] is our \u201cbest guess\u201d for the true difference in\u00a0 proportions, [latex] p_{1} - p_{2}[\/latex]. Use technology to obtain the sample statistic and state\u00a0 what this value means in the context of the data.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Question 4<\/h3>\r\nThough we have a \u201cbest guess\u201d for the difference in proportions of regular voters\u00a0 between the two groups, we expect there is some variability associated with that\u00a0 guess. In other words, if we calculated the difference in proportions of regular voters\u00a0 from two other random samples of 3,594 eligible voters with a major party affiliation\u00a0 and 2,242 eligible voters without an affiliation, we would expect to get a different (yet\u00a0 probably close) value of [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex] than we did in the previous question. When certain conditions apply (more on those later), the sampling distribution tells\u00a0 us three things about the distribution of [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex]:\r\n
        \r\n \t
      1. For large samples, the distribution is normal.<\/li>\r\n \t
      2. The distribution has a mean of [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex], the true population difference.<\/li>\r\n \t
      3. The distribution has a standard deviation of [latex] \\sqrt{\\frac{p_{1}(1-p_{1})}{n_{1}} + \\frac{p_{2}(1-p_{2})}{n_{2}}}[\/latex]. This is the estimate of the sample-to-sample variability, the random variability we expect in\u00a0 [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex] if we take random samples of the same size repeatedly. In the formula,\u00a0 [latex] p_{1} - p_{2} [\/latex] are the true population proportions as previously stated, and [latex] n_{1} - n_{2} [\/latex] are the sample sizes for the group with a major party affiliation and the group\u00a0 without a major party affiliation, respectively.<\/li>\r\n<\/ol>\r\nBefore calculating a confidence interval, let\u2019s calculate the standard deviation.\u00a0 Similar to previous calculations, we will replace\u00a0[latex] p_{1} [\/latex] and\u00a0[latex] p_{2} [\/latex] in the formula with the\u00a0 respective sample proportions [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex]. The estimate is called the standard error.\r\n\r\nUse technology to obtain the standard error.\r\n\r\n<\/div>\r\n
        \r\n

        Question 5<\/h3>\r\nNow that we have our estimate of the difference in proportions and the standard\u00a0 error, let\u2019s calculate the confidence interval. The formula for the confidence interval\u00a0 is:\r\n\r\nEstimate\u00a0[latex] \\pm [\/latex] Margin of Error\r\n\r\n[latex] (\\hat{p_{1}} - \\hat{p_{2}}) \\pm z^{*} \\times \\sqrt{\\frac{\\hat{p_{1}}(1-\\hat{p_{1}})}{n_{1}} + \\frac{\\hat{p_{2}}(1-\\hat{p_{2}})}{n_{2}}} [\/latex]\r\n\r\nRecall from Question 2 that the estimate is the difference in the sample proportions. Let\u2019s break down the margin of error. The margin of error is the width of the confidence interval and is comprised to two parts:\r\n