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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 occur in this hypothesis test if it was concluded that the proportion of 18 to 29-year-olds who voted in the 2018 election was higher than 50%, when in fact it was not. Conversely, a Type II error would occur if it was concluded that the proportion of 18 to 29-year-olds who voted in the 2018 election was not higher than 50%, when in fact it was.

Step by step solution

01

Understanding Type I Error

A Type I error would occur if it was concluded that the proportion of 18 to 29-year-olds who voted in the 2018 election was higher than 50%, when in fact it was not. This could lead to incorrect assumptions or actions based on the belief that there was an increase in voter participation, when there was actually no significant change.
02

Understanding Type II Error

A Type II error would occur if it was concluded that the proportion of 18 to 29-year-olds who voted in the 2018 election was not higher than 50%, when in fact it was. This could lead to missed opportunities or incorrect actions based on the belief that there was no significant change in voter participation, when there was actually an increase.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error
In statistical hypothesis testing, a Type I error refers to the faulty rejection of a true null hypothesis. Imagine you're a researcher assessing voter participation among young individuals. You start with the assumption that the voter rate remains unchanged from a previous election, which is your null hypothesis.
In this scenario, a Type I error would be analogous to a false alarm: you enthusiastically report an increase in young voter turnout when, in actuality, it's unchanged. The implications can be significant, leading to misguided policies or campaigns based on this incorrect assumption that more young people are voting. To minimize Type I errors, researchers adjust the significance level, or the probability of making such an error, usually setting a strict threshold—often 5%—as the cutoff for claiming a finding is statistically significant.
This ensures that only strong evidence can refute the null hypothesis, leading to more confidence in the results of statistical tests.
Type II Error
Conversely, a Type II error occurs when a false null hypothesis isn't rejected. Using the voter participation example, this would mean you've incorrectly concluded there's no increase in the 18 to 29-year-old voter participation when, in fact, more young people have headed to the polls than before.
The ramifications of a Type II error can lead to complacency or a failure to recognize positive changes within the electorate, potentially missing out on strengthening strategies that successfully mobilize young voters. To decrease the risk of a Type II error, one could increase the sample size, enhancing the study's power to detect true differences in voter participation rates if they indeed exist.
Voter Participation Statistics
Voter participation statistics play a critical role in understanding the dynamics of an electorate, particularly among different age groups. With the claim that 50% of eligible 18 to 29-year-olds voted in the 2016 election, statisticians are interested in identifying trends or shifts in subsequent elections.
Accurate measurement of these statistics is important for tailoring political strategies and allocating campaign resources. For instance, if statistics show increasing voter participation among young people, political parties might focus on issues that resonate with this demographic. To ensure the reliability of these statistics, robust sampling methods and thorough analyses must be employed, accounting for various factors that could influence voter turnout.
Statistical Significance
Statistical significance is a measure of whether the results observed in a study are likely to be due to chance or if they reflect a true effect. It is often determined by a p-value, which quantifies the probability of obtaining results at least as extreme as those observed, under the assumption that the null hypothesis is true.
If the p-value falls below a predetermined threshold, such as 0.05, the results are deemed statistically significant, and the null hypothesis is rejected. For the 2018 election voting rates scenario, if the statistical tests produce a p-value less than 0.05, we might conclude that there is indeed an increase in young voter turnout. Nevertheless, it's essential to interpret significance with caution, as it doesn't measure the size of an effect nor does it guarantee that the results are practically meaningful or free from errors.

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Most popular questions from this chapter

Historically (from about 2001 to 2014 ), \(57 \%\) of Americans believed that global warming is caused by human activities. A March 2017 Gallup poll of a random sample of 1018 Americans found that 692 believed that global warming is caused by human activities. a. What percentage of the sample believed global warming was caused by human activities? b. Test the hypothesis that the proportion of Americans who believe global warming is caused by human activities has changed from the historical value of \(57 \%\). Use a significance level of \(0.01\). c. Choose the correct interpretation: i. In 2017 , the percentage of Americans who believe global warming is caused by human activities is not significantly different from \(57 \%\). ii. In 2017 , the percentage of Americans who believe global warming is caused by human activities has changed from the historical level of \(57 \%\).

Pew Research conducts polls on social media use. In \(2012,66 \%\) of those surveyed reported using Facebook. In 2018 , \(76 \%\) reported using Facebook. a. Assume that both polls used samples of 100 people. Do a test to see whether the proportion of people who reported using Facebook was significantly different in 2012 and 2018 using a \(0.01\) significance level. b. Repeat the problem, now assuming the sample sizes were both 1500 . (The actual survey size in 2018 was \(1785 .\).) c. Comment on the effect of different sample sizes on the p-value and on the conclusion.

Suppose we are testing people to see whether the rate of use of seat belts has changed from a previous value of \(88 \%\). Suppose that in our random sample of 500 people we see that 450 have the seat belt fastened. Which of the following figures has the correct p-value for testing the hypothesis that the proportion who use seat belts has changed? Explain your choice.

Suppose you wanted to test the claim that the majority of U.S. voters are satisfied with the government response to the opioid crisis. State the null and alternative hypotheses you would use in both words and symbols.

St. Louis County is \(24 \%\) African American. Suppose you are looking at jury pools, each with 200 members, in St. Louis County. The null hypothesis is that the probability of an African American being selected into the jury pool is \(24 \%\). a. How many African Americans would you expect on a jury pool of 200 people if the null hypothesis is true? b. Suppose pool A contains 40 African American people out of 200 , and pool B contains 26 African American people out of 200 . Which will have a smaller p-value and why?

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