Question

The Wall Street Journal reports 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 $16,642. Assume the standard deviation is

σ = $2,400.

(a)

What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $200 of the population mean for each of the following sample sizes: 20, 60, 150, and 500? (Round your answers to four decimal places.)

sample size n = 20 sample size n = 60 sample size n = 150 sample size n = 500

(b)

What is the advantage of a larger sample size when attempting to estimate the population mean?

A larger sample increases the probability that the sample mean will be within a specified distance of the population mean.A larger sample lowers the population standard deviation.    A larger sample increases the probability that the sample mean will be a specified distance away from the population mean.A larger sample has a standard error that is closer to the population standard deviation.

Answer 1

a) For n = 20

   P(16442 < < 16842)

= P(-0.373 < Z < 0.373)

= P(Z < 0.373) - P(Z < -0.373)

= 0.6454 - 0.3546

= 0.2908

For n = 60

   P(16442 < < 16842)

= P(-0.645 < Z < 0.645)

= P(Z < 0.645) - P(Z < -0.645)

= 0.7405 - 0.2595

= 0.4810

For n = 150

P(16442 < < 16842)

= P(-1.021 < Z < 1.021)

= P(Z < 1.021) - P(Z < -1.021)

= 0.8464 - 0.1536

= 0.6928

For n = 500

P(16442 < < 16842)

= P(-1.863 < Z < 1.863)

= P(Z < 1.863) - P(Z < -1.863)

= 0.9688 - 0.0312

= 0.9376

b) A larger sample increases the probability that the sample mean will be within a specified distance of the population mean. A larger sample lowers the population standard deviation.