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Chapter 8: Problem 45

According to the Brookings Institution, \(50 \%\) of eligible 18 - to 29 -year- old voters voted in the 2016 election. Suppose we were interested in whether the proportion of voters in this age group who voted in the 2018 election was higher. Describe the two types of errors we might make in conducting this hypothesis test.

Short Answer

Expert verified
A Type I error would be to mistakenly infer that more people between the ages of 18 and 29 voted in 2018 compared to 2016 when it isn't true. On the other hand, a Type II error would occur if it was incorrectly concluded that there wasn't an increase in voter turnout among this age group in 2018 when there actually was an increase.

Step by step solution

01

Understanding the Scenario

Based on our goal, 'interested in whether the proportion of voters in this age group who voted in the 2018 election was higher', we label the hypothesis as follows: The null hypothesis (\(H_0\)): The proportion of voters aged 18-29 years was the same or lower in the 2018 election compared to 2016. The alternative hypothesis (\(H_A\)): The proportion of voters aged 18-29 years was higher in the 2018 election compared to 2016.
02

Identify Type I Error

A type I error occurs when the null hypothesis is true, but is incorrectly rejected in favor of the alternative hypothesis. In this context, a Type I error would be to conclude that the proportion of voters aged 18-29 who voted in 2018 was higher than in 2016 when it actually was the same or lower. This might result in unnecessary measures or actions taken to understand and address a non-existent increase in the voting trends of the specified age group.
03

Identify Type II Error

A type II error occurs when the null hypothesis is false, but is not rejected. This means we fail to accept the alternative hypothesis when it is the one that is true. Within this context, a Type II error would be failing to conclude that the proportion of voters aged 18-29 who voted in 2018 was higher than in 2016, when it actually did increase. This might result in a missed opportunity to identify and understand the factors responsible for the increased voter turnout in this age group, which may have been useful for future elections.

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Most popular questions from this chapter

A Gallup poll conducted in 2017 found that 648 out of 1011 people surveyed supported same-sex marriage. An NBC News/Wall Street Journal poll conducted the same year surveyed 1200 people and found 720 supported same-sex marriage. a. Find both sample proportions and compare them. b. Test the hypothesis that the population proportions are not equal at the \(0.05\) significance level.

A Quinnipiac poll conducted on February 20 , 2018 , found that 824 people out of 1249 surveyed favored stricter gun control laws. A survey conducted one week later on February 28 , 2018 , by National Public Radio found that 754 out of 1005 people surveyed favored stricter gun control laws. a. Find both sample proportions and compare them. b. Test the hypothesis that the population proportions are not equal at the \(0.05\) significance level. c. After conducting the hypothesis test, a further question one might ask is what is the difference between the two population proportions? Find a \(95 \%\) confidence interval for the difference between the two proportions and interpret it. How does the confidence interval support the hypothesis test conclusion?

An immunologist is testing the hypothesis that the current flu vaccine is less than \(73 \%\) effective against the flu virus. The immunologist is using a \(1 \%\) significance level and these hypotheses: \(\mathrm{H}_{\mathrm{o}}: p=0.73\) and \(\mathrm{H}_{\mathrm{a}}: p<0.73\). Explain what the \(1 \%\) significance level means in context.

Suppose you are testing someone to see whether he or she can tell butter from margarine when it is spread on toast. You use many bite-sized pieces selected randomly, half from buttered toast and half from toast with margarine. The taster is blindfolded. The null hypothesis is that the taster is just guessing and should get about half right. When you reject the null hypothesis when it is actually true, that is often called the first kind of error. The second kind of error is when the null is false and you fail to reject. Report the first kind of error and the second kind of error.

In 2017 the Pew Research Center conducted a survey on family-leave practices and attitudes. Respondents were asked to complete this sentence: "When a family member has a serious health condition, caregiver responsibilities ..." with choices being "mainly on women," "mainly on men," or "on both men and women equally." The percentages for each response are shown in the table below. For these age groups, the responses fell only into the two categories shown in the table. Assume a sample size of 1200 for each age group. $$\begin{array}{|lcc|}\hline \text { Age } & \begin{array}{l}\text { Fall mainly } \\ \text { on women }\end{array} & \begin{array}{l} \text { Fall equally on } \\\\\text { men and women }\end{array} \\\\\hline 30-49 & 60 \% & 40 \% \\\50-64 & 62 \% & 38 \%\end{array}$$ Can we conclude that there is a difference in the proportion of people aged 30 to 49 and aged 50 to 64 who feel the primary caregiver responsibility falls on women? Use a significance level of \(0.05\).
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