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 $17,801. Assume that the standard deviation is σ=$2546. 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 $212 of the population mean for each of the following sample sizes: 30,50 , 100, and 400 ? Round your answers to four decimals.

n=30
n=50
n=100
n=400
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b. What is the advantage of a larger sample size when attempting to estimate the population mean? Round your answer: to four decimals. A larger sample the probability that the sample mean will be within a specified distance of the population mean. In this instance, the probability of being within ±212 of μ ranges from sample of size 30 to for a sample of size 400 .

Answer 1

The probabilities for different sample sizes can be calculated using the z-table. A larger sample size increases the probability of the sample mean being within a specified distance of the population mean.

Step-by-step explanation:

The probability can be calculated using the z-table.

For a sample size of 30:

Z-score = (212 - 17801) / (2546 / sqrt(30)) = -1.0341.

Using the z-table, the probability is 0.1492.

For a sample size of 50:

Z-score = (212 - 17801) / (2546 / sqrt(50)) = -1.0416.

Using the z-table, the probability is 0.1478.

For a sample size of 100:

Z-score = (212 - 17801) / (2546 / sqrt(100)) = -1.0472.

Using the z-table, the probability is 0.1462.

For a sample size of 400:

Z-score = (212 - 17801) / (2546 / sqrt(400)) = -1.0492.

Using the z-table, the probability is 0.1458.

A larger sample size increases the probability that the sample mean will be within a specified distance of the population mean. In this case, the probability of being within ±212 of the population mean increases as the sample size increases from 30 to 400.