Question

For Exercises 7 through \(27,\) perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Coupon Use In today's economy, everyone has become savings savvy. It is still believed, though, that a higher percentage of women than men clip coupons. A random survey of 180 female shoppers indicated that 132 clipped coupons while 56 out of 100 men did so. At \(\alpha=0.01,\) is there sufficient evidence that the proportion of couponing women is higher than the proportion of couponing men? Use the \(P\) -value method.

Short answer

There is sufficient evidence at \(\alpha = 0.01\) to conclude that more women clip coupons than men.

Step-by-step solution

State the Hypotheses

We define the null hypothesis ( H_0 ) and the alternative hypothesis ( H_1 ) as follows: - H_0: p_1 = p_2 (The proportion of women who clip coupons is equal to the proportion of men who do.) - H_1: p_1 > p_2 (The proportion of women who clip coupons is greater than the proportion of men who do.) Here, p_1 represents the proportion of women who clip coupons, and p_2 represents the proportion of men who clip coupons. The claim we are testing is that the proportion of couponing women is higher than men.

Find the Critical Value(s)

Since we are using a significance level of \(\alpha = 0.01\) and the alternative hypothesis indicates a one-tailed test (right-tailed), we need to find the critical value for zi that corresponds to \(1 - \alpha\) in the standard normal distribution.The critical value z_c at \(\alpha = 0.01\) for a right-tailed test is approximately 2.33.

Compute the Test Value

First, we calculate the pooled sample proportion:\[\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{132 + 56}{180 + 100} = \frac{188}{280} \approx 0.6714\]Then, we calculate the standard error for the difference in proportions:\[SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.6714 \cdot (1 - 0.6714) \left(\frac{1}{180} + \frac{1}{100}\right)} \approx 0.0604\]Next, we compute the sample proportions \(p_1\) and \(p_2\):\[p_1 = \frac{132}{180} \approx 0.7333\quad \text{and} \quad p_2 = \frac{56}{100} = 0.56\]Finally, we find the test statistic zi using:\[z_i = \frac{p_1 - p_2}{SE} = \frac{0.7333 - 0.56}{0.0604} \approx 2.87\]

Make the Decision

We compare the test statistic \(z_i \approx 2.87\) to the critical value \(z_c = 2.33\). Since \(z_i\) is greater than \(z_c\), we reject the null hypothesis H_0.This decision indicates that at a significance level of 0.01, there is enough evidence to support the claim that a higher proportion of women clip coupons than men.

Summarize the Results

The hypothesis test yielded a test statistic that exceeds the critical value, leading to the rejection of the null hypothesis in favor of the alternative hypothesis. At \(\alpha = 0.01\), the evidence is sufficient to conclude that a higher percentage of women clip coupons compared to men. This supports the claim that women are more likely than men to use coupons.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement or assumption that there is no effect or difference, and it acts as a default position to be tested. In this exercise, the null hypothesis \( H_0 \) is: \( p_1 = p_2 \), meaning the proportion of women and men who clip coupons is equal. Here, \( p_1 \) represents the population proportion of women who use coupons, while \( p_2 \) stands for that of men.
The null hypothesis is crucial because it sets the stage for statistical testing. It's like the "status quo" we assume is true until evidence suggests otherwise. Rejection of \( H_0 \) indicates that there is a statistically significant effect or difference. This hypothesis is always tested under the assumption of its truth, making our statistical analysis an effort to find strong evidence against it.
All hypothesis tests begin by assuming that \( H_0 \) is true, and then statistics are calculated to determine whether to reject it. This decision is based on the probability of observing the data assuming the null hypothesis is correct.

Alternative Hypothesis

The alternative hypothesis, marked as \( H_1 \), is the statement you want to test, and it contradicts the null hypothesis. For this exercise, the alternative hypothesis is \( H_1: p_1 > p_2 \). This hypothesis suggests that the proportion of women who clip coupons is greater than that of men.
Unlike the null hypothesis, the alternative hypothesis is what you expect to support with your statistical evidence. It aims to demonstrate a significant effect or relationship. The formulation of the alternative hypothesis depends on the research question and is derived from the claim made at the beginning of the analysis.
In one-tailed tests, the alternative hypothesis might claim that one population parameter is either greater than or less than the other, as in our case with more women clipping coupons than men. This directional focus helps determine the critical region that will lead to rejecting the null hypothesis.

Critical Value

A critical value is a threshold that determines the boundary of the critical region in hypothesis testing. It is a value from a probability distribution, which in the case of a standard normal distribution and a right-tailed test at \( \alpha = 0.01 \), is approximately 2.33.
Critical values are vital because they provide the comparison point for the test statistic, which is evaluated under the assumption of the null hypothesis being true. The selection of a critical value depends on the significance level \( \alpha \) which indicates the probability of rejecting the null hypothesis when it is true (Type I error).
In one-tailed tests like this one, you only look at one side of the distribution to determine whether the test statistic falls into the critical region. If the test statistic exceeds the critical value in a right-tailed test, the null hypothesis is rejected.

Test Statistic

The test statistic is a value calculated from sample data that is used to evaluate the truth of the null hypothesis. In our exercise, the test statistic \( z_i \) was computed as approximately 2.87. The formula for the test statistic in this problem involves the difference between sample proportions of women and men clipping coupons, divided by the standard error of the difference.
The test statistic serves as the bridge between the sample data and the decision-making process in hypothesis testing. It translates the observed data into a standard form allowing comparison with critical values.
A test statistic that falls in the critical region indicates that the sample provides enough evidence to reject \( H_0 \). It's essentially an index of how far the sample data is from what would be expected under the null hypothesis. The test statistic is compared against the critical value to make a decision about the null hypothesis.

One-tailed Test

A one-tailed test is a type of hypothesis test where the region of rejection is on only one side of the sampling distribution. The one-tailed test is utilized when the research hypothesis is directional, suggesting that one parameter is greater than or less than another.
In this exercise, we conduct a right-tailed test to determine whether a greater proportion of women clip coupons than men. Given the alternative hypothesis \( H_1: p_1 > p_2 \), we place our entire focus on whether the observed test statistic is sufficiently large to reject the null hypothesis.
One-tailed tests increase the test's power to detect an effect in one specific direction. However, they require strong justification since they ignore the possibility of an effect in the other direction. By concentrating the critical region on one side, one-tailed tests allow for potentially smaller critical values and thus may lead to more frequent rejections of \( H_0 \) under real differences, as seen in our exercise.