176k views
1 vote
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

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 .

User Syuaa SE
by
7.9k points

1 Answer

7 votes

Final answer:

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.

User Sergiomafra
by
8.0k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories