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. Political Science: Voters A random sample of \(n_{1}=288\) voters registered in the state of California showed that 141 voted in the last general election. A random sample of \(n_{2}=216\) registered voters in the state of Colorado showed that 125 voted in the most recent general election. (See reference in Problem \(31 .\) Do these data indicate that the population proportion of voter turnout in Colorado is higher than that in California? Use a \(5 \%\) level of significance.

Step by step solution

01

Define the Significance Level and Hypotheses

We are testing whether the proportion of voter turnout in Colorado is higher than in California, with a significance level of \( \alpha = 0.05 \). The null hypothesis \( H_0 \) is that the population proportions are equal: \( p_1 = p_2 \). The alternative hypothesis \( H_a \) is that the proportion in Colorado is greater: \( p_1 < p_2 \).

02

Check Requirements and Assumptions

We will use the normal approximation to the binomial distribution to test the difference in proportions since both sample sizes are large. We assume that the samples are independent and randomly selected, and the sampling distribution of the difference in sample proportions is approximately normal.

03

Calculate the Sample Test Statistic

First, we compute the sample proportions: \( \hat{p}_1 = \frac{141}{288} \approx 0.4896 \) for California and \( \hat{p}_2 = \frac{125}{216} \approx 0.5787 \) for Colorado. The test statistic for the difference in proportions is given by: \[ Z = \frac{(\hat{p}_2 - \hat{p}_1) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]where \( \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \) is the pooled sample proportion, \( x_1 = 141 \) and \( x_2 = 125 \). Calculating gives \( \hat{p} \approx 0.5271 \), and substituting the values yields \( Z \approx 1.993 \).

04

Find the P-value

The \( P \)-value is determined from the standard normal distribution for \( Z = 1.993 \). Using standard normal distribution tables, or software, we find \( P(Z > 1.993) \approx 0.0233 \).

05

Decision and Conclusion

Since the \( P \)-value (0.0233) is less than the significance level (0.05), we reject the null hypothesis. This suggests that the data provides sufficient evidence to conclude that the voter turnout proportion in Colorado is higher than in California at the \( 5\% \) significance level.

06

Interpretation in Context

Interpreting in context, the results suggest that a higher proportion of registered voters participated in the last general election in Colorado compared to California. This conclusion is statistically significant at the \( 5\% \) level of significance.

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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 by \( \alpha \), helps us determine the threshold for rejecting the null hypothesis. The most commonly used significance level is 5\% or \( \alpha = 0.05 \), which indicates that there is a 5\% risk of concluding that there is an effect when there is none. This level of significance serves as a cut-off point where we decide whether to accept or reject the null hypothesis.

If the \( P \)-value, which measures the probability of observing the test results under the null hypothesis, is less than \( \alpha \), we reject the null hypothesis. If it is greater, we fail to reject the null hypothesis. Deciding on the significance level before testing ensures that we have a systematic and unbiased approach to determining statistical significance.

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is the statement we aim to test. It usually represents a statement of no effect or no difference. In our exercise, the null hypothesis suggests that the voter turnout proportion in Colorado is equal to that in California. This is expressed mathematically as \( p_1 = p_2 \).

The reason we use a null hypothesis is that it provides a baseline or reference point against which we can measure any observed effects. By initially assuming that no effect exists, we can use statistical evidence to objectively determine whether we observe a genuine effect. It's important to note that rejecting the null hypothesis doesn't mean it is false, just that the data provides sufficient evidence against it under the chosen level of significance.

Alternate Hypothesis

While the null hypothesis indicates no effect or no difference, the alternate hypothesis represents what we suspect might be true instead. In the context of our voting example, the alternate hypothesis \( H_a \) suggests that the voter turnout proportion in Colorado is actually higher than in California. Mathematically, this is expressed as \( p_1 < p_2 \).

The purpose of formulating an alternate hypothesis is to have a different claim ready to accept if the null hypothesis is rejected. It usually aligns with the goals of the research, reflecting an assertion we are interested in providing evidence for. When the \( P \)-value indicates that the observed data are unlikely under the null hypothesis, we have a reason to support the alternate hypothesis.

P-value

The \( P \)-value is a critical component of hypothesis testing, providing the probability of observing the test data, or something more extreme, assuming the null hypothesis is true. It helps us decide the strength of evidence against the null hypothesis.

In our example, with a calculated \( P \)-value of approximately 0.0233, it implies that there is a 2.33\% probability the observed difference in voter turnout proportions is due to random variation alone. Since this \( P \)-value is lower than our significance level \( \alpha = 0.05 \), we have enough statistical evidence to reject the null hypothesis. A smaller \( P \)-value means stronger evidence against \( H_0 \), indicating a potentially significant result.

Standard Normal Distribution

The standard normal distribution, also known as the Z-distribution, is a crucial concept in hypothesis testing, particularly when working with large sample sizes. It is a normal distribution with a mean of 0 and a standard deviation of 1, denoted as \( Z \sim N(0,1) \). In hypothesis testing, it is used to calculate \( Z \)-scores, which help us measure the number of standard deviations a data point is from the mean.

When assessing differences in proportions, as in our exercise, using the standard normal distribution allows us to derive the \( Z \)-statistic. This statistic helps determine how far away our sample finding is from the null hypothesis. By referencing the standard normal distribution, we can find the \( P \)-value associated with our \( Z \)-statistic, enabling us to make informed decisions about hypothesis validity. Understanding this distribution is essential for anyone working with hypothesis testing as it underlies many statistical tests.