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 $16,675 . Assume that the standard deviation is $2,645 . 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 $194 of the population mean for each of the following sample sizes: 30, 50, 100, and 400? Round your answers to four decimals.

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 -increasesdecreasesItem 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 +-194 of  ranges from  for a sample of size 30 to  for a sample of size 400 .

Answer 1

The probability of sample mean within $194 of the population mean is,

Converting to standard normal distribution,

For n = 30 ,

Using the standard normal distribution table,

For n = 50 ,

Using the standard normal distribution table,

For n = 100 ,

Using the standard normal distribution table,

For n = 400 ,

Using the standard normal distribution table,

As the sample size increases more and data are approaching towards population means (also known as Central Limit Theorem)

A larger sample increases, the probability that the sample mean will be within a specified distance of the population mean