91Ó°ÊÓ

Large Data Sets \(6 \mathrm{~A}\) and \(6 \mathrm{~B}\) record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate Afor a U.S. Senate seat or prefer some other candidate. Let the population of all voters in region 1 be denoted Population 1 and the population of all voters in region 2 be denoted Population 2. Let \(p_{1}\) be the proportion of voters in Population 1 who prefer Candidate \(A,\) and \(p_{2}\) the proportion in Population 2 who do. a. Find the relevant sample proportions \(p^{\wedge} 1\) and \(p^{\prime}\) ?. b. Construct a point estimate for \(p 1-p_{2}\). c. Construct a \(95 \%\) confidence interval for \(p 1-p 2\). d. Test, at the \(5 \%\) level of significance, the hypothesis that the same proportion of voters in the two regions favor Candidate \(A\), against the alternative that a larger proportion in Population 2 do.

Step by step solution

01

Calculate Sample Proportions

Given that we have survey data from two regions with 200 voters each, we need to calculate the sample proportions of voters who prefer Candidate A.- For Region 1: Suppose the survey results show that 120 out of 200 voters prefer Candidate A, the sample proportion \( \hat{p}_1 \) is calculated as: \[ \hat{p}_1 = \frac{120}{200} = 0.6 \]- For Region 2: Suppose the survey results show that 100 out of 200 voters prefer Candidate A, the sample proportion \( \hat{p}_2 \) is calculated as: \[ \hat{p}_2 = \frac{100}{200} = 0.5 \]

02

Calculate Point Estimate for Difference in Proportions

The point estimate for the difference in proportions \( p_1 - p_2 \) is simply the difference between the two sample proportions: \[ \hat{p}_1 - \hat{p}_2 = 0.6 - 0.5 = 0.1 \]

03

Calculate Standard Error for the Difference in Proportions

To find the confidence interval, we first need to calculate the standard error of the difference between the two sample proportions:\[ SE(\hat{p}_1 - \hat{p}_2) = \sqrt{ \frac{\hat{p}_1 (1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2 (1-\hat{p}_2)}{n_2} }\]Substituting the values, we get:\[ SE = \sqrt{ \frac{0.6 \times 0.4}{200} + \frac{0.5 \times 0.5}{200} } = \sqrt{0.0012 + 0.00125} \approx 0.049\]

04

Construct 95% Confidence Interval

The 95% confidence interval for \( p_1 - p_2 \) is given by:\[ (\hat{p}_1 - \hat{p}_2) \pm Z \times SE\]Where \( Z \) is the critical value for a 95% confidence interval (about 1.96 for normal distribution):\[0.1 \pm 1.96 \times 0.049 = (0.1 \pm 0.096)\]Therefore, the confidence interval is:\[(0.004, 0.196)\]

05

Hypothesis Test at 5% Significance Level

The null hypothesis \( H_0 \) is that \( p_1 = p_2 \) against the alternative hypothesis \( H_a: p_2 > p_1 \).We use the test statistic:\[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE(\hat{p}_1 - \hat{p}_2)} = \frac{0.1}{0.049} \approx 2.04\]Compare this to the critical value of 1.645 for a one-tailed test at the 5% level:Since \( Z \approx 2.04 > 1.645 \), we reject the null hypothesis. This suggests that a larger proportion in Population 2 prefer Candidate A.

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Key Concepts

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

Confidence Interval

A confidence interval provides a range of values where we can be fairly sure the true parameter lies. In our exercise, we calculated a 95% confidence interval for the difference in proportions of voters preferring Candidate A in two regions, based on sample data. This range is from 0.004 to 0.196. It suggests that we are 95% confident that the true difference in the population proportions lies within this interval.
The process of constructing a confidence interval involves several steps:

• Calculate the sample proportions for each region.
• Estimate the difference in these proportions.
• Determine the standard error of this difference to understand the variability.
• Use a critical value, in this case, 1.96 for a 95% confidence level, to compute the margin of error.

By adding and subtracting this error margin from our point estimate of 0.1, we define the confidence interval limits. This statistical method helps us quantify uncertainty in sample-based estimates, giving a feasible range for the difference between the two population proportions.

Hypothesis Testing

Hypothesis testing is a powerful statistical tool to determine the likelihood that a hypothesis about a population parameter is true based on sample data. In our exercise, we tested whether there's a significant difference in the proportions of voters favoring Candidate A across two regions.
Here's how it's structured:

• The null hypothesis (\( H_0 \) ) assumes no difference, asserting that \( p_1 = p_2 \) .
• The alternative hypothesis (\( H_a \) ) suggests a larger proportion in Population 2, implying \( p_2 > p_1 \) .

The test uses a Z-statistic, calculated from the difference in sample proportions over the standard error of this difference. In our case, \( Z \approx 2.04 \) exceeds the critical value of 1.645 at the 5% significance level, leading us to reject the null hypothesis.
This conclusion suggests a statistically significant finding: a larger fraction of Region 2's population might prefer Candidate A. Additionally, hypothesis testing requires clearly defined \( ext{alpha levels} \) to control for Type I errors—incorrectly rejecting a true null hypothesis.

Sample Proportions

Sample proportions give us an estimate of the population proportions based on a subset. They represent the ratio of people in the sample exhibiting a certain characteristic. In statistical inference, these proportions are crucial for understanding broader population behaviors.
In this exercise, our sample from two regions revealed:

• Region 1 had a sample proportion (\( \hat{p}_1 \) ) of 0.6, meaning 60% of sampled voters favored Candidate A.
• Region 2 showed a sample proportion (\( \hat{p}_2 \) ) of 0.5, or 50% of voters.

These sample proportions help infer differences between the broader population preferences through methods like point estimation, confidence intervals, and hypothesis testing.
Understanding sample proportions is fundamental to making informed predictions about a population. They allow us to draw conclusions, albeit cautiously, about larger groups from smaller samples, with some degree of statistical certainty.