91Ó°ÊÓ

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. According to the U.S. Bureau of Labor Statistics, approximately equal numbers of men and women are engaged in sales and related occupations. Although that may be true for total numbers, perhaps the proportions differ by industry. A random sample of 200 salespersons from the industrial sector indicated that 114 were men, and in the medical supply sector, 80 of 200 were men. At the 0.05 level of significance, can we conclude that the proportion of men in industrial sales differs from the proportion of men in medical supply sales?

Step by step solution

01

State Hypotheses and Identify the Claim

To determine whether the proportions of men differ between industrial sales and medical supply sales, we state our hypotheses. - Null Hypothesis \( (H_0) \): \( p_1 = p_2 \), where \( p_1 \) is the proportion of men in industrial sales, and \( p_2 \) is the proportion of men in medical supply sales.- Alternative Hypothesis \( (H_1) \): \( p_1 eq p_2 \). The claim here is the alternative hypothesis that the proportions differ.

02

Find the Critical Value(s)

This is a two-tailed test since we are testing whether the proportions are different. At the 0.05 level of significance, we look for the critical z-value in a standard normal distribution table. For a 0.05 significance level, the critical z-values are approximately -1.96 and 1.96.

03

Compute the Test Value

- First, calculate the sample proportions: - For industrial sales: \( \hat{p}_1 = \frac{114}{200} = 0.57 \) - For medical supply sales: \( \hat{p}_2 = \frac{80}{200} = 0.40 \)- Calculate the pooled sample proportion: \( \hat{p} = \frac{114 + 80}{200 + 200} = \frac{194}{400} = 0.485 \)- Now, calculate the standard error (SE): \[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.485 \times 0.515 \times \left(\frac{1}{200} + \frac{1}{200}\right)} = 0.0498 \]- Compute the test statistic Z: \[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.57 - 0.40}{0.0498} \approx 3.37 \]

04

Make the Decision

Compare the computed test statistic with critical values. Here, the calculated Z-value is 3.37, which is greater than the critical value of 1.96. Therefore, we reject the null hypothesis \(H_0\).

05

Summarize the Results

Since the null hypothesis is rejected, there is enough statistical evidence at the 0.05 level of significance to support the claim that the proportion of men in industrial sales differs from the proportion of men in medical supply sales.

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Key Concepts

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

Null Hypothesis

The concept of the null hypothesis is foundational in hypothesis testing. In our context, a null hypothesis, denoted as \(H_0\), proposes that there is no difference between two proportions. For the salespersons example, the null hypothesis is \(p_1 = p_2\), where \(p_1\) is the proportion of men in industrial sales and \(p_2\) is in medical supply sales. The role of the null hypothesis is to provide a standard of comparison for evaluating if there is an effect or difference. This hypothesis presumes "no effect" or "no difference," meaning any observed variation is due to chance.

Alternative Hypothesis

Opposite of the null, the alternative hypothesis challenges the status quo by suggesting a difference exists. It is denoted as \(H_1\) in hypothesis testing. In the example at hand, the alternative hypothesis indicates \(p_1 eq p_2\), implying a difference in the proportions of men between industrial and medical supply sales. It includes the actual claim we want to test. If the evidence suggests that the null hypothesis is not true, we support the alternative hypothesis. This principle makes it vital in drawing conclusions from the data.

Critical Value

The critical value is like a threshold in hypothesis testing. It's the point past which we reject the null hypothesis. In a two-tailed test like ours, we're interested if the test statistic falls below or above this critical range. At a significance level of 0.05, we look up these values in a z-table, finding critical values of approximately -1.96 and 1.96. These values delimit the regions on both ends of the normal curve where, if our test statistic lands, it leads us to reject the null hypothesis.

Test Statistic

The test statistic is what we calculate from our sample data to perform the hypothesis test. It tells us how far our sample statistic (like a sample proportion) deviates from the null hypothesis's expectation in standard units (e.g., standard errors). For our example, we computed a Z-score of 3.37. This step uses the sample data and an equation to translate the data outcomes into a single value. When this "test statistic" is far from zero, it suggests a strong departure from the null hypothesis, hinting at supporting the alternative hypothesis.

Statistical Significance

Understanding statistical significance is crucial. It indicates whether the test results are likely due to chance or show a real effect. In hypothesis testing, if our test statistic falls in the critical region, it is deemed "statistically significant." In our exercise, the computed Z-feel of 3.37 exceeds the critical boundaries of -1.96 and 1.96, meaning the results are not likely due to random chance. At a 0.05 significance level, we're only willing to accept a 5% risk that the differences we observe (in men's proportions) are due to random chance, thus affirming the evidence for a difference between the two sectors.