91Ó°ÊÓ

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 (BusinessWeek, 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?

Step by step solution

01

Identifying the Proportion and Sample Size

For parts (a), (b), and (c), identify the sample proportion \( \bar{p} \) and sample size \( n \).- Part (a): \( \bar{p} = 0.53 \), \( n = 1500 \).- Part (b): \( \bar{p} = 0.31 \), \( n = 1500 \).- Part (c): \( \bar{p} = 0.05 \), \( n = 1500 \).

02

Finding the Z-score for the Confidence Interval

For each part, determine the z-score for a 95% confidence interval. The z-score for a 95% confidence level is 1.96.

03

Calculating the Standard Error for Each Proportion

The standard error (SE) is calculated using the formula: \[ SE = \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \]- Part (a): \( SE = \sqrt{\frac{0.53(1-0.53)}{1500}} \approx 0.0128 \)- Part (b): \( SE = \sqrt{\frac{0.31(1-0.31)}{1500}} \approx 0.0117 \)- Part (c): \( SE = \sqrt{\frac{0.05(1-0.05)}{1500}} \approx 0.0055 \)

04

Calculating the Margin of Error for Each Proportion

The margin of error (ME) is calculated using the formula:\[ ME = Z \times SE \]- Part (a): \( ME = 1.96 \times 0.0128 \approx 0.025 \)- Part (b): \( ME = 1.96 \times 0.0117 \approx 0.023 \)- Part (c): \( ME = 1.96 \times 0.0055 \approx 0.011 \)

05

Developing the Confidence Interval for Each Proportion

Use the formula for the confidence interval (CI):\[ CI = (\bar{p} - ME, \bar{p} + ME) \]- Part (a): \( CI = (0.505, 0.555) \)- Part (b): \( CI = (0.287, 0.333) \)- Part (c): \( CI = (0.039, 0.061) \)

06

Comparing the Margins of Error

- Part (a) Margin of Error: 0.025- Part (b) Margin of Error: 0.023- Part (c) Margin of Error: 0.011The margin of error is related to the proportion \( \bar{p} \) as it is affected by \( SE \), which includes \( \bar{p}(1-\bar{p}) \). The proportion closest to 0.5 generally leads to a larger margin of error.

07

Understanding the Planning Value

The planning value \( p^{*} = 0.50 \) is often used when calculating sample sizes as it maximizes the product \( p(1-p) \), leading to the largest possible margin of error. This ensures a conservative estimate, capturing the widest range of potential values.

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Key Concepts

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

Proportion Estimation

When we talk about proportion estimation, we're trying to understand the fraction of a whole that exhibits a certain characteristic. Let's say you surveyed 1,500 wealthy individuals to find out how many experienced a significant loss in their portfolio, and you found that 53% did. This percentage, or proportion, is a snapshot of your sample, represented statistically as \( \bar{p} \). Proportion estimation aims to extend this snapshot to the broader population from which the sample was drawn.
By calculating the sample proportion, we develop insights into the population proportion with a level of certainty. This is especially useful when data stems from surveys or polls.
To estimate the proportion in a population:

• Start with your sample proportion \( \bar{p} \), which is the ratio of favorable outcomes to the total sample size.
• In the example above, \( \bar{p} = 0.53 \) for part (a), meaning that 53% of the respondents saw a portfolio loss.
• Then use statistical methods to extend this sample finding to the entire population, within a known error margin.

Establishing such connections is essential for making predictions and informed decisions about larger groups.

Margin of Error

The margin of error provides a range within which we believe the true population proportion lies. It's a crucial aspect of estimating proportions because it accounts for the natural variability inherent in sample data. When you calculate the margin of error, you're basically setting boundaries around your sample proportion to represent a possible spread of the true population proportion. This concept gives you a clearer picture of how much trust you should place in the results from your sample.
To compute the margin of error, you need two things: the standard error and the z-score for your desired confidence level (often 1.96 for a 95% confidence interval).

• The Standard Error (SE) is calculated using the formula: \[ SE = \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \]
• The Margin of Error (ME) is then given by: \[ ME = Z \times SE \]

This range describes just how much the sample proportion might differ from the actual population proportion. A wider margin of error suggests more uncertainty, while a narrower one indicates more confidence in the results; that’s why understanding the margin of error's relationship with \( \bar{p} \) and the sample size is vital.

Sample Size

The sample size, often denoted by \( n \), plays a pivotal role in determining the accuracy of our proportion estimations. In statistics, the sample size is the number of observations used to calculate estimates for the population parameters. Larger sample sizes typically yield more reliable results, with narrower confidence intervals and smaller margins of error. This makes them crucial for achieving a high level of accuracy in statistical analyses.
But why is a large sample size so beneficial? Here are three key reasons:

• With more data points, there's a greater chance of capturing the diversity and variation present in the population, leading to more representative results.
• A larger sample size reduces the impact of outliers or anomalies on the overall findings.
• It allows for more precise estimates by decreasing the standard error, which directly affects the margin of error.

Therefore, when planning a study, the goal is often to decide on a sample size that balances precision with practicality. The larger the \( n \), the more trustworthy your confidence intervals will be, reflecting a true glimpse of the population being studied.