91Ó°ÊÓ

Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding \(z\) or \(t\) value as appropriate. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom \(d . f .\) not in the Student's \(t\) table, use the closest \(d . f .\) that is smaller. In some situations, this choice of \(d . f .\) may increase the \(P\) -value a small amount and therefore produce a slightly more "conservative" answer. Art: Politics Do you prefer paintings in which the people are fully clothed? This question was asked by a professional survey group on behalf of the National Arts Society (see reference in Problem 30 ). A random sample of \(n_{1}=59\) people who are conservative voters showed that \(r_{1}=45\) said yes. Another random sample of \(n_{2}=62\) people who are liberal voters showed that \(r_{2}=36\) said yes. Does this indicate that the population proportion of conservative voters who prefer art with fully clothed people is higher than that of liberal voters? Use \(\alpha=0.05\)

Step by step solution

01

State the Hypotheses and Significance Level

We are testing if the proportion of conservative voters who prefer fully clothed people in paintings is higher than that of liberal voters. Thus, we set up the hypotheses as follows:- Null Hypothesis (H_0): \( p_1 - p_2 = 0 \)- Alternative Hypothesis (H_a): \( p_1 - p_2 > 0 \)The level of significance is \( \alpha = 0.05 \).

02

Check the Requirements and Calculate the Test Statistic

To decide the appropriate sampling distribution and compute the test statistic, we assume the proportions follow a normal distribution because both sample sizes are large enough (both greater than 30).Calculate sample proportions:\[ \hat{p}_1 = \frac{45}{59}, \quad \hat{p}_2 = \frac{36}{62} \]Pooled sample proportion:\[ \hat{p} = \frac{r_1 + r_2}{n_1 + n_2} = \frac{45 + 36}{59 + 62} \]Standard error (SE):\[ SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \]Test statistic:\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]

03

Determine the P-Value

Using the calculated z-score from Step 2, look up the P-value from standard normal distribution tables (or use statistical software). The P-value represents the probability of observing a test statistic as extreme as, or more extreme than, what was observed, under the assumption that the null hypothesis is true. Illustrate this by sketching the standard normal distribution, marking the calculated z-value, and shading the area beyond this point (to the right, since this is a right-tailed test).

04

Make a Decision

Compare the calculated P-value with the significance level \( \alpha = 0.05 \):- If \( P \leq \alpha \), reject the null hypothesis.- If \( P > \alpha \), fail to reject the null hypothesis.Based on the P-value obtained from Step 3, make the appropriate decision regarding the null hypothesis.

05

Interpret the Conclusion

Based on the statistical decision in Step 4, interpret in the context of the problem:If the null hypothesis was rejected, it suggests that there is significant evidence at the \( \alpha=0.05 \) level to conclude that the proportion of conservative voters who prefer fully clothed paintings is indeed higher than that of liberal voters.If the null hypothesis was not rejected, it suggests that there is not enough evidence to support the claim that conservative voters prefer fully clothed paintings more than liberal voters at the \( \alpha=0.05 \) level.

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Key Concepts

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

Significance Level

In hypothesis testing, the significance level, denoted as \( \alpha \), is a threshold used to determine whether the observed data are statistically significant. A common choice for \( \alpha \) is 0.05, as used in our exercise. This means we allow a 5% probability of rejecting the null hypothesis when it is actually true. The significance level reflects our tolerance for type I error, which is the risk of making a false positive conclusion.Choosing \( \alpha = 0.05 \) is a balance between detecting a real effect and minimizing the chances of falsely claiming there is an effect when there isn't one. If the P-value in our test is less than or equal to \( \alpha \), we reject the null hypothesis, suggesting that our findings are statistically significant at the 5% level. Remember, a smaller \( \alpha \) would require more compelling evidence against the null hypothesis to be considered significant, whereas a larger \( \alpha \) may lead to more frequent false positives.

Sampling Distribution

The sampling distribution is a critical concept in hypothesis testing. It describes the probability distribution of a given statistic based on a random sample. In our case, where we're comparing proportions, the sampling distribution of the difference between sample proportions \( \hat{p}_1 - \hat{p}_2 \) is used.With large enough sample sizes (usually \( n > 30 \) is considered sufficient), we can assume the sampling distributions of sample proportions are approximately normally distributed due to the Central Limit Theorem. Therefore, the Z-distribution is employed for our test. This is because sample sizes of 59 and 62 are sufficiently large, allowing for the normal approximation. The sampling distribution assists in calculating the test statistic and, subsequently, the P-value, facilitating the evaluation of whether the observed difference is likely due to chance.

P-Value

The P-value is the probability of observing test results at least as extreme as the observed results, assuming that the null hypothesis is true. It helps us determine the statistical significance of our hypothesis test.In our exercise, we calculate a test statistic (z-value) to help find the P-value. By looking at the standard normal distribution, this P-value corresponds to the probability of the test statistic being further from zero in the direction defined by the alternative hypothesis.If the P-value is less than or equal to \( \alpha \), the result is considered statistically significant, and we reject the null hypothesis. A larger P-value implies weaker evidence against the null hypothesis, suggesting that our observed results could easily occur under the null hypothesis.

Test Statistic

A test statistic is a standardized value used to determine the degree to which the sample data are consistent with the null hypothesis. In our example, we utilize the Z-test for comparing two proportions since our sample sizes are large.The formula is:\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]The standard error (SE) in this formula is derived from the pooled sample proportion. The test statistic measures how far the observed sample statistic is from the null hypothesis value, in units of standard error. The further the observed statistic is from the expected value, the more likely it is that the null hypothesis is false.By converting this into a Z-score, we can reference standard normal distribution tables to find the P-value, which assists in making informed decisions in hypothesis testing.

Null Hypothesis and Alternative Hypothesis

In hypothesis testing, the null hypothesis (\( H_0 \)) and alternative hypothesis (\( H_a \)) are fundamental concepts that outline what we aim to test. In our example, the null hypothesis states there is no difference in the proportions of conservative and liberal voters preferring fully clothed art: \( H_0: p_1 - p_2 = 0 \).The alternative hypothesis captures what we suspect is true, which is that the proportion of conservative voters preferring this art style is higher: \( H_a: p_1 - p_2 > 0 \).These hypotheses are mutually exclusive, meaning that evidence against the null hypothesis is considered support for the alternative hypothesis. The goal of our test is to gather sufficient evidence to either reject the null hypothesis, thereby supporting the alternative hypothesis, or fail to reject it to maintain the status quo. Restating the hypotheses makes it easier to understand the statistical results and what they imply in the context of our research question.