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 $16,028 . Assume that the standard deviation is=$2997 . Use z-table.

a. What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $209 of the population mean for each of the following sample sizes:30 ,50 ,100 ,400 and ? Round your answers to four decimals.

n=30 _____

n=50 ______

n=100_____

n=400_____

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 - Select your answer -(increases/decreases)Item 5 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 +/- 209 of  ranges from______ for a sample of size 30 to_______ for a sample of size 400 .

Answer 1

a)

for n=30,

for normal distribution z score =(X-μ)/σx
here mean=       μ=	16028
std deviation   =σ=	2997.0000
sample size       =n=	30
std error=σx̅=σ/√n=	547.1748

probability =

P(15819<X<16237)

=

P(-0.38<Z<0.38)=

0.6480-0.3520=

0.2960

as above:

n	probability
30	0.2960
50	0.3758
100	0.5160
400	0.8354

b)

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 +/- 209 of  ranges from 0.2960 for a sample of size 30 to 0.8354 for a sample of size 400 .