A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with net worth of \(\$ 1\) million or more provided a variety of statistics on wealthy people (Business Week, September 22,2003 ). The previous three-year period had been bad for the stock market, which motivated some of the questions asked. a. The survey reported that \(53 \%\) of the respondents lost \(25 \%\) or more of their portfolio value over the past three years. Develop a \(95 \%\) confidence interval for the proportion of wealthy people who lost \(25 \%\) or more of their portfolio value over the past three years. b. The survey reported that \(31 \%\) of the respondents feel they have to save more for retirement to make up for what they lost. Develop a \(95 \%\) confidence interval for the population proportion. c. Five percent of the respondents gave \(\$ 25,000\) or more to charity over the previous year. Develop a \(95 \%\) confidence interval for the proportion who gave \(\$ 25,000\) or more to charity. d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to \(\bar{p} ?\) When the same sample is being used to estimate a variety of proportions, which of the proportions should be used to choose the planning value \(p^{*} ?\) Why do you think \(p^{*}=.50\) is often used in these cases?

Short Answer

Expert verified
The 95% confidence intervals for the proportions of wealthy people who (a) lost 25% or more of their portfolio value, (b) feel they have to save more for retirement, and (c) gave $25,000 or more to charity are (0.5049, 0.5551), (0.2860, 0.3340), and (0.0388, 0.0612), respectively. The margin of error is larger when the sample proportion is closer to 0.5, which increases the uncertainty of the estimate. In many cases, a planning value of 0.5 is used because it represents the largest possible margin of error, providing a more conservative estimate.

Step by step solution

01

Part (a): Confidence interval for losing 25% or more of portfolio value

First, let's calculate the 95% confidence interval for the proportion of wealthy people who lost 25% or more of their portfolio value. We are given that 53% of the respondents reported this, and the sample size is 1500. 1. Identify the sample proportion (\(\hat{p}_1\)) and sample size (n): \(\hat{p}_1 = 0.53\) and \(n = 1500\). 2. Calculate the standard error (SE): \(SE = \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{n}} = \sqrt{\frac{0.53 (1 - 0.53)}{1500}} \approx 0.0128\). 3. Find the critical value (Z) for a 95% confidence interval: \(Z = 1.96\). 4. Calculate the margin of error (ME): \(ME = Z \times SE = 1.96 \times 0.0128 \approx 0.0251\). 5. Calculate the confidence interval: \((\hat{p}_1 - ME, \hat{p}_1 + ME) = (0.53 - 0.0251, 0.53 + 0.0251) = (0.5049, 0.5551)\). So, the 95% confidence interval for the proportion of wealthy people who lost 25% or more of their portfolio value over the past three years is (0.5049, 0.5551).
02

Part (b): Confidence interval for feeling the need to save more for retirement

Next, let's calculate the 95% confidence interval for the proportion of wealthy people who feel they have to save more for retirement. We are given that 31% of the respondents reported this. 1. Identify the sample proportion (\(\hat{p}_2\)) and sample size (n): \(\hat{p}_2 = 0.31\) and \(n = 1500\). 2. Calculate the standard error (SE): \(SE = \sqrt{\frac{\hat{p}_2 (1 - \hat{p}_2)}{n}} = \sqrt{\frac{0.31 (1 - 0.31)}{1500}} \approx 0.0123\). 3. Find the critical value (Z) for a 95% confidence interval: \(Z = 1.96\). 4. Calculate the margin of error (ME): \(ME = Z \times SE = 1.96 \times 0.0123 \approx 0.0240\). 5. Calculate the confidence interval: \((\hat{p}_2 - ME, \hat{p}_2 + ME) = (0.31 - 0.0240, 0.31 + 0.0240) = (0.2860, 0.3340)\). So, the 95% confidence interval for the proportion of wealthy people who feel they have to save more for retirement is (0.2860, 0.3340).
03

Part (c): Confidence interval for giving $25,000 or more to charity

Finally, let's calculate the 95% confidence interval for the proportion of wealthy people who gave $25,000 or more to charity. We are given that 5% of the respondents reported this. 1. Identify the sample proportion (\(\hat{p}_3\)) and sample size (n): \(\hat{p}_3 = 0.05\) and \(n = 1500\). 2. Calculate the standard error (SE): \(SE = \sqrt{\frac{\hat{p}_3 (1 - \hat{p}_3)}{n}} = \sqrt{\frac{0.05 (1 - 0.05)}{1500}} \approx 0.0057\). 3. Find the critical value (Z) for a 95% confidence interval: \(Z = 1.96\). 4. Calculate the margin of error (ME): \(ME = Z \times SE = 1.96 \times 0.0057 \approx 0.0112\). 5. Calculate the confidence interval: \((\hat{p}_3 - ME, \hat{p}_3 + ME) = (0.05 - 0.0112, 0.05 + 0.0112) = (0.0388, 0.0612)\). So, the 95% confidence interval for the proportion of wealthy people who gave $25,000 or more to charity is (0.0388, 0.0612).
04

Part (d): Comparison of margin of error and relationship with sample proportion

Now let's compare the margin of error for the three confidence intervals: - Part (a): ME1 = 0.0251 - Part (b): ME2 = 0.0240 - Part (c): ME3 = 0.0112 The margin of error is larger when the sample proportion is closer to 0.5. This is because the variance is maximized when the true proportion is 0.5, which leads to increased uncertainty in the estimate. When determining a planning value for the proportion, it is often advisable to choose the proportion with the largest margin of error, since this will provide the most conservative estimate of the true proportion. In many cases, a planning value of 0.5 is used because it represents the worst-case scenario in terms of the largest possible margin of error, providing a more conservative estimate and assuring a higher level of confidence in the resulting interval.

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