Solve And Show All Work: Political Polls Typically Sample Ra
Solve And Show All Workpolitical Polls Typically Sample Randomly From
Political polls typically sample randomly from the U.S. population to investigate the percentage of voters who favor some candidate or issue. The number of people polled is usually on the order of 1,000. Suppose that one such poll asks voters how they feel about the President’s handling of the crisis in the financial markets. The results show that 575 out of 1,280 people polled say they either “approve” or “strongly approve” of the President’s handling of this matter. Based on these data, find a 95% confidence interval estimate for the proportion of the entire voter population who “approve” or “strongly approve” of the President’s handling of the crisis. Now, consider an alternative scenario where the sample proportion is the same as before, but the sample size is doubled—that is, 1,150 out of 2,560—and analyze how this would affect the confidence interval.
Paper For Above instruction
In statistical analysis, especially in public opinion polling, estimating the proportion of a population that supports a certain candidate or issue relies heavily on sampling methods and confidence interval calculations. The primary goal is to infer the true proportion of the entire population based on a representative sample. In this paper, we analyze data from a political poll to estimate the proportion of voters who approve or strongly approve of the President’s handling of a financial crisis and explore the implications of increasing the sample size on the confidence interval.
The initial data consist of 575 favorable responses out of 1,280 voters surveyed, yielding a sample proportion \( \hat{p}_1 = \frac{575}{1280} \approx 0.4492 \). To construct a 95% confidence interval for the true population proportion \( p \), we use the standard formula based on the normal approximation to the binomial distribution, which is appropriate given the sample size and the expected distribution of the responses.
The formula for the confidence interval is:
\[
CI = \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
where \( \hat{p} \) is the sample proportion, \( n \) is the sample size, and \( Z_{\alpha/2} \) is the critical value from the standard normal distribution corresponding to the desired confidence level. For a 95% confidence interval, \( Z_{0.025} \approx 1.96 \).
Substituting the known values:
\[
\hat{p}_1 = 0.4492, \quad n = 1280
\]
we compute the standard error (SE):
\[
SE = \sqrt{\frac{0.4492 \times (1 - 0.4492)}{1280}} \approx \sqrt{\frac{0.4492 \times 0.5508}{1280}} \approx \sqrt{\frac{0.2472}{1280}} \approx \sqrt{0.000193} \approx 0.0139
\]
Multiplying by the critical value:
\[
Margin\ of\ error = 1.96 \times 0.0139 \approx 0.0272
\]
Therefore, the 95% confidence interval is approximately:
\[
(0.4492 - 0.0272, \quad 0.4492 + 0.0272) \approx (0.4220, \quad 0.4764)
\]
This means we are 95% confident that between approximately 42.20% and 47.64% of the entire voter population approve or strongly approve of the President’s handling of the crisis.
In the second scenario, the sample size increases to 2,560 with the same sample proportion \( \hat{p}_2 = \frac{1150}{2560} \approx 0.4492 \). The standard error becomes:
\[
SE_{new} = \sqrt{\frac{0.4492 \times 0.5508}{2560}} \approx \sqrt{\frac{0.2472}{2560}} \approx \sqrt{0.0000966} \approx 0.0098
\]
The margin of error with the larger sample:
\[
1.96 \times 0.0098 \approx 0.0192
\]
And the new confidence interval:
\[
(0.4492 - 0.0192, \quad 0.4492 + 0.0192) \approx (0.4300, \quad 0.4684)
\]
Notably, increasing the sample size narrows the confidence interval from approximately (0.4220, 0.4764) to (0.4300, 0.4684). This demonstrates that larger samples provide more precise estimates of the population proportion, reducing the margin of error and increasing the reliability of the inference.
In conclusion, increasing the sample size from 1,280 to 2,560 significantly improves the precision of the estimate, evidenced by the narrower confidence interval. This characteristic highlights the importance of sample size in statistical sampling and confidence interval construction, particularly in political polling where accurate estimates influence public opinion and decision-making processes.
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