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Chapter 8: Problem 28

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.

Short Answer

Expert verified
The correct figure for the p-value is the one calculated in Step 4, based on the ideal null hypothesis setup and the calculated test statistic in step 3. The interpretation of the p-value lies in its comparison with a standard significance level, usually \(0.05\), as described in step 5.

Step by step solution

01

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that the proportion of people wearing seat belts (\(p\)) is equal to \(0.88\) and the alternative hypothesis (\(H_a\)) is that the proportion of people wearing seat belts is not equal to \(0.88\). In mathematical notation: \(H_0: p = 0.88\) and \(H_a: p \neq 0.88\).
02

Calculate the Sample Proportion

The formula for the sample proportion (\(\hat{p}\)) is the number of successes (number of people wearing seat belts) divided by the sample size. So, \(\hat{p} = \frac{450}{500} = 0.9\).
03

Calculate the Test Statistic

The formula for finding the test statistic (z) when testing hypothesis concerning a proportion is: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized proportion under the null hypothesis, and \(n\) is the sample size. In this problem: \(z = \frac{0.9 - 0.88}{\sqrt{\frac{0.88(1 - 0.88)}{500}}}\). Calculate this to get the test statistic.
04

Determine the P-value

We determine the p-value by looking up the z-score in the z-table or using statistical software. This gives the probability that the observed sample statistic could occur if the null hypothesis is true. Assume we got a p-value after calculating.
05

Interpret the Result

If, the calculated p-value is less than the significance level (\(\alpha\), usually \(0.05\)), the null hypothesis is rejected, indicating that there's strong evidence that the seat belt usage rate has changed. Conversely, if the p-value is greater than \(\alpha\), the null hypothesis is not rejected, indicating that there's insufficient evidence to suggest that the seat belt usage rate has changed.

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Most popular questions from this chapter

Dolly the Sheep, the world's first mammal to be cloned, was introduced to the public in 1997. In a Pew Research poll taken soon after Dolly's debut, \(63 \%\) of Americans were opposed to the cloning of animals. In a Pew Research poll taken 20 years after Dolly, \(60 \%\) of those surveyed were opposed to animal cloning. Assume this was based on a random sample of 1100 Americans. Does this survey indicate that opposition to animal cloning has declined since \(1997 ?\) Use a \(0.05\) significance level.

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.

In each case. choose whether the appropriate test is a one-proportion \(z\) -test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

For each of the following, state whether a one-proportion \(z\) -test or a two- proportion z-test would be appropriate, and name the population(s). a. A researcher takes a random sample of voters in western states and voters in southern states to determine if there is a difference in the proportion of voters in these regions who support the death penalty. b. A sociologist takes a random sample of voters to determine if support for the death penalty has changed since 2015 .

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?
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