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# A random sample of 24 Mason students was collected and each student was asked, among other things,

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Investigation 1: Appropriateness of Inference: Price of a Haircut

For the following scenario, answer the questions below.  Please note, do not conduct inference in this problem; just answer each question.

A random sample of 24 Mason students was collected and each student was asked, among other things, the total cost of their last hair service (including cuts, styling, etc.).  The researcher does not have any information about the population from which the sample was collected.  The data set is called HairPrice.

If we attempt to conduct statistical inference using the collected sample, what is the parameter of interest?  Use the correct symbol and describe the parameter in context in one sentence.

Check the specific conditions necessary to consider conducting inference using theory-based methods using the t-distribution.  There are three to consider: (1) Was a random sample collected; (2) Is the population where the sample comes from normal; and (3) Is the sample size greater than or equal to 30?  Answer each of these questions in one sentence.

What could be checked using the sample data if our sample size is less than 30?  Answer this question in one sentence.

Depending on your answer to part (a), construct one or two frequency histograms in Rguroo.  Remember to properly title and label the graph(s).  Copy and paste this graph (or these graphs) into your document.

Describe the shape of the histogram(s) in one sentence.

Depending on your answer to part (a), construct one or two horizontal boxplots in Rguroo.  Remember to properly title and label the graph(s).  Copy and paste this graph (or these graphs) into your document.

Does the boxplot (or do the boxplots) show any outliers?  Answer this question in one sentence and identify any outliers if they are present.

Considering your answers to parts (e) and (g), is theory-based inference appropriate in this case?  If you respond “yes,” provide a reason for your response.  If you respond “no,” state the reason why not and present another possibility if the researcher still wanted to conduct statistical inference.  Use one or two complete sentences in your response.

Investigation 2: Earnings among Voters

A political scientist wondered if there is a significant difference between the proportion of Democrats and Republicans earning over \$100,000.  To obtain the data, she used the National Election Pool.  The NEP is a consortium of major new networks (ABC, CBS, CNN, and NBC) that pools together resources to gather voting and exit poll data from a random sample of voters.  On Election Day, November 8, 2022, exit poll data showed that among a random sample of 774 voters, 408 were registered as Republican and 366 were registered as Democrat.  Of the 408 Republican voters, 216 earn over \$100,000 a year.  Of the 366 Democrat voters, 162 earn over \$100,000 a year. Use  = 0.05.

Define the population parameter using context and symbols in one complete sentence.

State the hypotheses using the political scientist’s claim.

Check the specific conditions necessary to consider conducting inference using theory-based (or distribution based) methods.  There are two to consider: (1) Was the data collected randomly from the population; and (2) Are there at least ten successes and failures in each group?  Answer each of these questions in one sentence and show that condition (2) is true or false using calculation to obtain the failures (note the successes are given).

Calculate and label the two sample proportions separately and round the values to four decimal places.  Next, calculate the difference between these sample proportions by subtracting (Republican – Democrat).  Type all of these calculations and label each of them.

Calculate the pooled proportion estimate needed in the calculation of the standard error of the test statistic.  Type this calculation and round to four decimal places.

Calculate the test statistic value using your proportions obtained in parts (d) and (e) and type your work.  Round your test statistic value to three decimal places.

Obtain your p-value using your test statistic calculated in part (f) in StatKey using Theoretical Distributions → Normal.  Copy and paste the image of the standard normal distribution and type the value of the p-value below the image.

Verify your test statistic and p-value using Rguroo.  Go to Analytics → Analysis → Proportion Inference → Two Populations.  See the image below to fill in each box correctly.  Then, click the Test of Hypothesis tab and choose Large Sample z under method.  Finally, correctly set your alternative hypothesis and significance level and click Preview.  Copy and paste only the output and table displayed under the title “Two Population Proportion Test of Hypothesis”

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