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Chapter 8: Problem 61

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.

Short Answer

Expert verified
A smaller p-value is obtained when the sample proportions are far apart. This is because a smaller p-value provides strong evidence against the null hypothesis, and a larger difference in sample proportions implies a larger observed effect, which in turn lowers the p-value.

Step by step solution

01

Understanding of p-value

The p-value is a measure of the likelihood that the observed data would occur given that the null hypothesis is true. The smaller the p-value, the stronger the evidence against the null hypothesis.
02

Explaining Hypothesis Testing Scenario

In a two-sided alternative hypothesis, the test is checking for the possibility of the relationship in both directions. Here, the null hypothesis is that the two sample proportions are the same.
03

Understanding the correlation between p-value and observed effect

When the sample proportions are farther apart, there is stronger evidence against the null hypothesis. This indicates a larger observed effect, which translates to a smaller p-value.

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Most popular questions from this chapter

A 2018 Gallup poll of 2228 randomly selected U.S. adults found that \(39 \%\) planned to watch at least a "fair amount" of the 2018 Winter Olympics. In \(2014,46 \%\) of U.S. adults reported planning to watch at least a "fair amount." a. Does this sample give evidence that the proportion of U.S. adults who planned to watch the 2018 Winter Olympics was less than the proportion who planned to do so in 2014 ? Use a \(0.05\) significance level. b. After conducting the hypothesis test, a further question one might ask is what proportion of all U.S. adults planned to watch at least a "fair amount" of the 2018 Winter Olympics. Use the sample data to construct a \(90 \%\) confidence interval for the population proportion. How does your confidence interval support your hypothesis test conclusion?

According to a 2017 AAA survey, \(35 \%\) of Americans planned to take a family vacation (a vacation more than 50 miles from home involving two or more immediate family members. Suppose a recent survey of 300 Americans found that 115 planned on taking a family vacation. Carry out the first two steps of a hypothesis test to determine if the proportion of Americans planning a family vacation has changed. Explain how you would fill in the required entries in the figure for # of success, # of observations, and the value in \(\mathrm{H}_{0}\).

The label on a can of mixed nuts says that the mixture contains \(40 \%\) peanuts. After opening a can of nuts and finding 22 peanuts in a can of 50 nuts, a consumer thinks the proportion of peanuts in the mixture differs from \(40 \%\). The consumer writes these hypotheses: \(\mathrm{H}_{0}: \mathrm{p} \neq 0.40\) and \(\mathrm{H}_{\mathrm{a}}: \mathrm{p}=0.44\) where \(p\) represents the proportion of peanuts in all cans of mixed nuts from this company. Are these hypotheses written correctly? Correct any mistakes as needed.

A study is done to see whether a coin is biased. The alternative hypothesis used is two-sided, and the obtained \(z\) -value is 1 . Assuming that the sample size is sufficiently large and that the other conditions are also satisfied, use the Empirical Rule to approximate the p-value.

About \(30 \%\) of the population in Silicon Valley, a region in California, are between the ages of 40 and 65, according to the U.S. Census. However, only \(2 \%\) of the 2100 employees at a laid-off man's former Silicon Valley company are between the ages of 40 and \(65 .\) Lawyers might argue that if the company hired people regardless of their age, the distribution of ages would be the same as though they had hired people at random from the surrounding population. Check whether the conditions for using the one-proportion z-test are met.
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