91Ó°ÊÓ

In the United States, historically, \(40 \%\) of registered voters are Republican. Suppose you obtain a simple random sample of 320 registered voters and find 142 registered Republicans. (a) Consider the hypotheses \(H_{0}: p=0.4\) versus \(H_{1}: p>0.4\). Explain what the researcher would be testing. Perform the test at the \(\alpha=0.05\) level of significance. Write a conclusion for the test. (b) Consider the hypotheses \(H_{0}: p=0.41\) versus \(H_{1}: p>0.41\). Explain what the researcher would be testing. Perform the test at the \(\alpha=0.05\) level of significance. Write a conclusion for the test. (c) Consider the hypotheses \(H_{0}: p=0.42\) versus \(H_{1}: p>0.42\). Explain what the researcher would be testing. Perform the test at the \(\alpha=0.05\) level of significance. Write a conclusion for the test. (d) Based on the results of parts (a)-(c), write a few sentences that explain the difference between "accepting" the statement in the null hypothesis versus "not rejecting" the statement in the null hypothesis.

Step by step solution

01

- Determine Hypotheses for Part (a)

Identify the null and alternative hypotheses.Null Hypothesis: \(H_{0}: p=0.4\)Alternative Hypothesis: \(H_{1}: p>0.4\)

02

- Set Significance Level for Part (a)

Set the significance level for the test: \(\alpha=0.05\)

03

- Compute Sample Proportion for Part (a)

Calculate the sample proportion \(\hat{p}\).Given 142 Republicans out of 320 voters, \(\hat{p} = \frac{142}{320} ≈ 0.44375\).

04

- Compute Test Statistic for Part (a)

Use the test statistic formula for proportions:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]where \(\hat{p} = 0.44375\), \(p_0 = 0.4\), and \(n = 320\)\[z = \frac{0.44375 - 0.4}{\sqrt{\frac{0.4 \times 0.6}{320}}} ≈ 1.594\]

05

- Determine p-value and Conclusion for Part (a)

Find the p-value corresponding to the z-score.The p-value for \(z ≈ 1.594\) is approximately 0.055.Since p-value (0.055) > \(\alpha\) (0.05), do not reject \(H_{0}\).Conclusion: There is not enough evidence to support that the proportion of Republican voters is greater than 40%.

06

- Determine Hypotheses for Part (b)

Identify the null and alternative hypotheses.Null Hypothesis: \(H_{0}: p=0.41\)Alternative Hypothesis: \(H_{1}: p>0.41\)

07

- Compute Test Statistic for Part (b)

Use the test statistic formula:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]where \(\hat{p} = 0.44375\), \(p_0 = 0.41\), and \(n = 320\)\[z = \frac{0.44375 - 0.41}{\sqrt{\frac{0.41 \times 0.59}{320}}} ≈ 1.260\]

08

- Determine p-value and Conclusion for Part (b)

Find the p-value corresponding to the z-score.The p-value for \(z ≈ 1.260\) is approximately 0.104.Since p-value (0.104) > \(\alpha\) (0.05), do not reject \(H_{0}\).Conclusion: There is not enough evidence to support that the proportion of Republican voters is greater than 41%.

09

- Determine Hypotheses for Part (c)

Identify the null and alternative hypotheses.Null Hypothesis: \(H_{0}: p=0.42\)Alternative Hypothesis: \(H_{1}: p>0.42\)

10

- Compute Test Statistic for Part (c)

Use the test statistic formula:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]where \(\hat{p} = 0.44375\), \(p_0 = 0.42\), and \(n = 320\)\[z = \frac{0.44375 - 0.42}{\sqrt{\frac{0.42 \times 0.58}{320}}} ≈ 0.917\]

11

- Determine p-value and Conclusion for Part (c)

Find the p-value corresponding to the z-score.The p-value for \(z ≈ 0.917\) is approximately 0.180.Since p-value (0.180) > \(\alpha\) (0.05), do not reject \(H_{0}\).Conclusion: There is not enough evidence to support that the proportion of Republican voters is greater than 42%.

12

- Interpret Results

Discuss the difference between 'accepting' versus 'not rejecting' the null hypothesis.'Accepting' the null hypothesis suggests it is true, while 'not rejecting' means there is not sufficient evidence to disprove it. In statistics, conclusions should be framed as 'not rejecting' rather than 'accepting' to avoid overconfidence in the null hypothesis.

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Key Concepts

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

Null Hypothesis

The null hypothesis, denoted as \(H_0\), represents a default assumption that there is no effect or no difference in a population. For example, in our exercise, the null hypothesis states that the proportion of Republican voters is equal to a historical value, such as 40%. This is expressed as \(H_0: p = 0.4\). When performing hypothesis testing, the aim is to test whether there is enough evidence against this default assumption. If we find sufficient evidence, we may reject the null hypothesis. If not, we fail to reject it. Think of the null hypothesis as the 'status quo' we are trying to challenge.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_1\) or \(H_a\), is what you might conclude when there is enough evidence to reject the null hypothesis. It represents a new claim you're testing, suggesting some effect or difference. In our case, the alternative hypothesis proposes that the actual proportion of Republican voters is greater than the historical proportion. If our null hypothesis is \(H_0: p = 0.4\), the alternative hypothesis would be \(H_1: p > 0.4\). This is a one-sided or one-tailed test because we are only interested in deviations in one direction (more than 40%). It is essential because rejecting the null hypothesis in favor of the alternative hypothesis can lead to new insights and decisions.

P-Value

The p-value measures the strength of evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. In simpler terms, a lower p-value indicates stronger evidence against \(H_0\). For example, if the calculated p-value is 0.03, this means there is a 3% chance that the observed data (or something more extreme) would occur if \(H_0\) were true. Generally, we compare the p-value to a significance level, \(\text{α}\), typically set at 0.05. If the p-value is less than \(\text{α}\), we reject the null hypothesis. In our exercise, p-values helped us determine whether the observed proportion of Republican voters significantly differed from the historical proportions.

Test Statistic

The test statistic is a standardized value used to decide whether to reject the null hypothesis. It compares your observed sample data to what you would expect under the null hypothesis. For proportions, the test statistic is often a z-score calculated by the formula:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]Where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized proportion, and \(n\) is the sample size. For instance, in Part (a) of the exercise, we calculated a z-score of approximately 1.594 when testing if \(p = 0.4\). This z-score is then used to find the corresponding p-value. The higher or lower the z-score from zero, the stronger the evidence against \(H_0\). Computing the z-score helps you quantify the difference between your sample data and the null hypothesis.

Significance Level

The significance level, denoted by \(\text{α}\), is the threshold used to determine whether to reject the null hypothesis. It represents the highest probability of Type I error you are willing to accept — that is, the risk of falsely rejecting \(H_0\). Common choices for \(\text{α}\) are 0.01, 0.05, and 0.10. In our exercise, a 0.05 significance level means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. If the p-value is less than \(\text{α}\), we reject \(H_0\). Otherwise, we do not reject \(H_0\). This level of significance is crucial in balancing the risks of making Type I and Type II errors (failing to reject a false null hypothesis). It's like setting the rule for decision-making in hypothesis testing.