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The paper "On the Nature of Creepiness" (New Ideas in Psychology [2016]: 10-15) describes a study to investigate what people think is "creepy." Each person in a sample of women and a sample of men were asked to do the following: Imagine a close friend of yours whose judgement you trust. Now imagine that this friend tells you that she or he just met someone for the first time and tells you that the person was creepy. The people in the samples were then asked whether they thought the creepy person was more likely to be a male or a female. Of the 1029 women surveyed, 980 said they thought it was more likely the creepy person was male, and 298 of the 312 men surveyed said they thought it was more likely the creepy person was male. Is there convincing evidence that the proportion of women who think the creepy person is more likely to be male is different from this proportion for men? For purposes of this exercise, you can assume that the samples are representative of the population of adult women and the population of adult men. Test the appropriate hypotheses using a significance level of 0.05

Step by step solution

01

State the Null and Alternative Hypotheses

Let p1 be the proportion of women who think the creepy person is more likely to be male, and p2 be the proportion of men who think the creepy person is more likely to be male. The null and alternative hypotheses are defined as follows: Null Hypothesis (H0): p1 = p2 Alternative Hypothesis (Ha): p1 ≠ p2

02

Compute the Test Statistic

To compute the test statistic, we use the formula for the difference of proportions: Z = \(\frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1} + \frac{\hat{p}(1-\hat{p})}{n_2}}}\) Where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions of women and men who think the creepy person is more likely to be male, n1 and n2 are the respective sample sizes, and \(\hat{p}\) is the combined proportion of both samples: \(\hat{p}_1 = \frac{980}{1029}\) \(\hat{p}_2 = \frac{298}{312}\) \(\hat{p} = \frac{980 + 298}{1029 + 312}\)

03

Calculate the Test Statistic and P-value

Using the given data and the formulas in Step 2, we can compute the test statistic: \(\hat{p}_1 = 0.9524\) \(\hat{p}_2 = 0.9551\) \(\hat{p} = 0.9531\) Z = \(\frac{(0.9524 - 0.9551) - 0}{\sqrt{\frac{0.9531(1-0.9531)}{1029} + \frac{0.9531(1-0.9531)}{312}}}\) Z ≈ -0.46 Now, we can find the P-value associated with this test statistic, which corresponds to the area in the tails of the standard normal distribution: P-value = 2 * P(Z < -0.46) = 2 * 0.3228 = 0.6456

04

Compare the P-value to the Significance Level

Now, we will compare the P-value to the significance level (α = 0.05): Since the P-value (0.6456) is greater than the significance level (0.05), we fail to reject the null hypothesis.

05

Make a Conclusion

Based on the hypothesis test, we fail to reject the null hypothesis that the proportion of women who think the creepy person is more likely to be male is the same as this proportion for men. Therefore, there is not convincing evidence to suggest that the proportions are different for men and women.

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Key Concepts

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

Understanding the Difference of Proportions

When you're dealing with the difference of proportions in hypothesis testing, you're comparing two separate groups to see if a particular trait is more common in one than the other. In this exercise, the goal is to compare the proportion of women and men who think a creepy person is more likely to be male. To do this, we calculate the sample proportions from each group:

• For women: 980 out of 1029 believe a creepy person is more likely male, so the proportion is \( \hat{p}_1 = \frac{980}{1029} \).
• For men: 298 out of 312 believe this, so the proportion is \( \hat{p}_2 = \frac{298}{312} \).

Once you have these, you determine if there's a significant difference between them by using a test statistic.

Formulating the Null Hypothesis

The null hypothesis is the starting point for any hypothesis test. It assumes that there is no effect or no difference—essentially, it's the status quo. For this particular exercise, the null hypothesis (H0) is that the proportion of women who think the creepy person is male is equal to the proportion of men who think the same: \[ H_0: p_1 = p_2 \] This hypothesis sets the ground for statistical testing. If your data provides enough evidence against this assumption, then we can consider alternative hypotheses.

Considering the Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis explores what you might expect if your initial hypothesis is not correct. In this case, our alternative hypothesis (Ha) states that the proportions are not equal: \[ H_a: p_1 eq p_2 \] This suggests there's some difference between how men and women perceive gender in creepy people. When testing hypotheses, rejecting the null hypothesis in favor of the alternative hypothesis indicates that such a difference exists. This is only done if there's sufficient statistical evidence to support the claim based on your data.

Calculating the P-value

A p-value helps you determine the significance of your results in hypothesis testing. In simpler terms, it's a way of quantifying the probability of finding your observed results—or something more extreme—if the null hypothesis is true. In this situation, you'd first calculate the test statistic using your sample data and then find the corresponding p-value from the standard normal distribution.For this problem:

• The combined sample proportion \( \hat{p} \) is found by pooling the data from both groups.
• The Z test statistic is calculated to determine how much \( \hat{p}_1 \) and \( \hat{p}_2 \) differ from each other under the null hypothesis.
• The p-value is then determined by finding the area under the curve beyond this Z value.

A p-value greater than your significance level (in this case, 0.05) suggests you don't have enough evidence to reject the null hypothesis, meaning you may conclude there's no strong overall difference between these gender proportions.