Question

The Wall Street Journal reported that 33% of taxpayers with adjusted gross incomes between $30,000 and $60,000 itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $16,928 . Assume that the standard deviation is $2,903.

(a) What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $151 of the population mean for each of the following sample sizes: 30, 50, 100, and 400? Round your answers to four decimals.

(b) What is the advantage of a larger sample size when attempting to estimate the population mean? Round your answers to four decimals.

A larger sample increases the probability that the sample mean will be within a specified distance of the population mean. In the automobile insurance example, the probability of being within (plus/minus) $151 of m ranges from (blank) for a sample of size 30 to (blank) for a sample of size 400.

Answer 1

Let X be the amount of deductions for this population of taxpayers.

X has a mean and standard deviation

a) Let represent the sample mean amount of deductions for a sample of size n

if n is greater than 30 or we know the population standard deviation, the central limit theorem says that has a normal distribution with mean and standard deviation (or called standard error of mean)

the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $151 of the population mean is same as the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean between (16928-151=16777) and (16928+151=17079)

When n=30, the standard error of mean is

the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean between 16777 and 17079 is

When n=50, the standard error of mean is

the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean between 16777 and 17079 is

When n=100, the standard error of mean is

the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean between 16777 and 17079 is

When n=400, the standard error of mean is

the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean between 16777 and 17079 is

ans: the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $151 of the population mean for each of the following sample sizes: 30, 50, 100, and 400 is

sample sizes (n)	Probability
30	0.2206
50	0.2886
100	0.3970
400	0.7016

b) the advantage of a larger sample size when attempting to estimate the population mean is

ans: A larger sample increases the probability that the sample mean will be within a specified distance of the population mean. In the itemized deductions (automobile insurance?) example, the probability of being within (plus/minus) $151 of m ranges from 0.2206 for a sample of size 30 to 0.7016 for a sample of size 400.