91Ó°ÊÓ

A study in the July 7,2009 , issue of \(U S A\) TODAY stated that the \(401(\mathrm{k})\) participation rate among U.S. employees of Asian heritage is \(76 \%\), whereas the participation rate among U.S. employees of Hispanic heritage is \(66 \%\). Suppose that these results were based on random samples of 100 U.S. employees from each group. a. Construct a \(95 \%\) confidence interval for the difference between the two population proportions. b. Using the \(5 \%\) significance level, can you conclude that the \(401(\mathrm{k})\) participation rates are different for all U.S. employees of Asian heritage and all U.S. employees of Hispanic heritage? Use the critical- value and \(p\) -value approaches. c. Repeat parts a and b for both sample sizes of 200 instead of 100 . Does your conclusion change in part b?

Step by step solution

01

Calculate Sample Proportions and Differences

First, calculate the sample proportions. For Asian: \(p_1 = 0.76\) and For Hispanic: \(p_2 = 0.66\). The difference in sample proportions is \(p_1 - p_2 = 0.76 - 0.66 = 0.10\).

02

Construct a 95% Confidence Interval for the Difference

Formula for confidence interval of difference in proportions is \[p_1 - p_2 \pm Z \sqrt{\frac{{p_1(1 - p_1)}}{{n_1}} + \frac{{p_2(1 - p_2)}}{{n_2}}}\]. Substituting, \(n_1 = 100, n_2 = 100, Z = 1.96\) (for 95% confidence), and the values of \(p_1\) and \(p_2\) from Step 1 into the formula to determine the confidence interval.

03

Perform Hypothesis Testing

The null hypothesis is that the proportions are equal, and the alternative hypothesis is that the proportions are not equal. Perform a Z test of hypothesis for the difference in proportions using the critical value and p-value approaches. If the calculated Z score is greater than the critical Z score, or the p-value is less than the predetermined significance level (0.05), reject the null hypothesis.

04

Repeat Steps 1-3 for a sample size of 200

Now repeat the same processes, but with the sample sizes \(n_1\) and \(n_2\) increased from 100 to 200. This involves recalculating the confidence interval using the increased sample size in the formula (Step 2) and rerunning the hypothesis test (Step 3).

05

Compare the Results

Compare the conclusions reached from the hypothesis tests and confidence intervals calculated with the sample sizes of 100 and 200. If these conclusions are different, then the sample size has an effect on the results of the analysis.

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Key Concepts

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

Understanding Hypothesis Testing

Hypothesis testing is a powerful statistical tool that helps you make inferences about a population based on sample data. It's like playing detective with data, where you try to determine if your findings are significant or just due to random chance.

When you conduct a hypothesis test, you start with two opposing statements:

• The null hypothesis (often denoted as \(H_0\)): This states that there is no effect or no difference. In our scenario, it suggests that the \(401(k)\) participation rates for Asian and Hispanic employees are the same.
• The alternative hypothesis (\(H_1\)): This states that there is an effect or a difference. Here, it implies that the participation rates are different for the two groups.

After setting these hypotheses, you go ahead and check your data. You determine if the evidence strongly supports either stance. If your analysis shows that the probability of the null hypothesis being true is too low, you reject \(H_0\) and lean toward \(H_1\).

Exploring Proportions

Proportions tell us how large one part of the whole group is. In the case of our study, proportions show the fraction of Asian and Hispanic employees participating in \(401(k)\) programs.

Here were the sample proportions:

• For Asian employees, the proportion \(p_1\) is \(0.76\) or 76%.
• For Hispanic employees, the proportion \(p_2\) is \(0.66\) or 66%.

Understanding these proportions lets us compare and quantify how many employees from each group are participating in the \(401(k)\) plan. This comparison is crucial in constructing confidence intervals and performing hypothesis tests.

The Importance of Sample Size

Sample size, often denoted with \(n\), can greatly influence the results of your analysis. It represents the number of observations or data points collected from a population.

When you increase the sample size, the estimate becomes more precise. A larger sample size can yield more reliable results because it better approximates the true population parameters. In our context:

• A sample size of 100 was initially used for both groups. With this size, the calculated difference in proportions seemed significant.
• When the sample size increased to 200, recalculating the confidence interval and hypothesis tests could lead to different results, underlining the impact of sample size on statistical conclusions.

Thus, choosing an adequate sample size is vital for robust and credible statistical inference.

Decoding Significance Level

The significance level, often represented by \(\alpha\), is a critical part of hypothesis testing. It tells us how strong the evidence must be to reject the null hypothesis.

In most studies:

• A common significance level is \(0.05\), or 5%. This means there's a 5% risk of concluding there is an effect when there truly isn't one.

This level helps set the critical threshold in determining whether our test results are statistically significant.

• If the p-value (a measure of the evidence against the null hypothesis) is less than \(\alpha\), you reject \(H_0\). It shows that the differences in proportions are unlikely due to chance alone.
• If the p-value is greater than \(\alpha\), you do not reject \(H_0\), suggesting there's not enough evidence to support that the participation rates differ significantly between the groups.

Selecting the right significance level is key to balancing the risk of incorrect conclusions.