A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with n

Question

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 (to 4 decimals).

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 (to 4 decimals).

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 (to 3 decimals).

Explanation / Answer

(a)

n = 1500    

p = 0.53    

% = 95    

Standard Error, SE = Ö{p(1 - p)/n} =    (0.53(1 - 0.53))/1500 = 0.012886686

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0128866856354404 = 0.02525744

Lower Limit of the confidence interval = P - width =     0.53 - 0.0252574397255529 = 0.50474256

Upper Limit of the confidence interval = P + width =     0.53 + 0.0252574397255529 = 0.55525744

The 95% confidence interval is [0.5047, 0.5553]

(b)

n = 1500    

p = 0.31    

% = 95    

Standard Error, SE = Ö{p(1 - p)/n} =    (0.31(1 - 0.31))/1500 = 0.011941524

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0119415241908225 = 0.02340496

Lower Limit of the confidence interval = P - width =     0.31 - 0.023404957334526 = 0.28659504

Upper Limit of the confidence interval = P + width =     0.31 + 0.023404957334526 = 0.33340496

The 95% confidence interval is [0.2866, 0.3334]

(c)

n = 1500    

p = 0.05    

% = 95    

Standard Error, SE = Ö{p(1 - p)/n} =    (0.05(1 - 0.05))/1500 = 0.005627314

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.00562731433871138 = 0.01102933

Lower Limit of the confidence interval = P - width =     0.05 - 0.0110293334335601 = 0.03897067

Upper Limit of the confidence interval = P + width =     0.05 + 0.0110293334335601 = 0.06102933

The 95% confidence interval is [0.039, 0.061]

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