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 $16,459 . Assume that the standard deviation is $2,496 . 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 of the $229 of 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 (either increases or decreases) when attempting to estimate the population mean? Round your answers to four decimals. A larger sample 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 +/- 229 of ranges from_____ for a sample of size 30 to_____ for a sample of size 400.

Answer 1

for n=30;

for normal distribution z score =(X-μ)/σx
here mean=       μ=	16459
std deviation   =σ=	2496.0000
sample size       =n=	30
std error=σx̅=σ/√n=	455.7052

probability =

P(16230<X<16688)

=

P(-0.5<Z<0.5)=

0.6915-0.3085=

0.3830

as above:

n	probability
30	0.383
50	0.4844
100	0.6424
400	0.9328

In the automobile insurance example, the probability of being within +/- 229 of ranges from 0.3830 for a sample of size 30 to 0.9328  for a sample of size 400.