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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
To determine the correct figure with the p-value, perform a hypothesis test. Check the test statistic calculated in step 3 and find its corresponding p-value in a standard normal distribution table.

Step by step solution

01

State the Hypotheses

The null hypothesis (\(H_0\)) is that the rate of use of seat belts is \(88 \%\), and the alternative hypothesis (\(H_1\)) is that the rate has changed (it could be either increased or decreased). In terms of proportions, this would mean \(H_0: p = 0.88\) and \(H_1: p \neq 0.88\).
02

Calculate the Sample Proportion

The proportion from the sample is calculated by dividing the number of successful outcomes (people wearing seat belts) by the total number of outcomes (total people). In this case, the sample proportion is \( \frac{450}{500} = 0.90\).
03

Perform the Hypothesis Test

Using a z test for proportions, calculate the test statistic. The formula for the test statistic in a one-proportion z-test is \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\). Here, \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion under null hypothesis, and \(n\) is the sample size. Plugging in the values, we find that the test statistic is \(Z = \frac{0.90 - 0.88}{\sqrt{\frac{0.88 (1 - 0.88)}{500}}}\).
04

Find the P-Value

The p-value associated with the calculated z-score should be found in a standard normal distribution table (also known as a z-table). Since this is a two-tailed test, the p-value will be equal to twice the value associated with the positive z-score (because of symmetry).

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