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). ( , )

D) Compare the margin of error for the interval estimates in parts (a), (b), and (c). The margin of error ____ as p-bar gets smaller.

Explanation / Answer

a) here p =0.53 ; n=1500

hence std error =(p(1-p)/n)1/2 =0.0129

for 95% CI, z =1.96

hence confidence interval =mean +/- z*std error =(0.505 ; 0.555)

b) for p=0.31

std error =0.012

confidence interval =0.287 ; 0.333

c) p=0.05

std error =0.0056

confidence interval =0.061 ; 0.939

d) as margin of error =z*std error

for part a) margin fo error =0.0253

for part B) margin of error =0.0234

for part c) margin of error =0.011

margin of error gets smaller as p gets smaller.

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