The Internal Revenue Service Is Studying The Category Of Cha

The Internal Revenue Service Is Studying The Category of Charitable Co

The Internal Revenue Service is examining the category of charitable contributions and has selected a sample of 18 tax returns from young couples aged 20 to 35 with an adjusted gross income over $100,000. Out of these 18 returns, 4 reported charitable contributions exceeding $1,000. Subsequently, three returns are chosen at random for a comprehensive audit. This scenario involves complex probability calculations, particularly using the hypergeometric distribution, to assess the likelihood of certain outcomes within the sample.

Part (a) requires calculating the probability that exactly one of the three audited returns reports a charitable deduction of more than $1,000. Since the sampling involves selections without replacement from a finite population with known success counts (returns with charitable contributions over $1,000), the hypergeometric distribution is appropriate. The formula for hypergeometric probability is:

P(X = k) = (C(K, k) * C(N - K, n - k)) / C(N, n)

where:

• N = total population size = 18
• K = total number of successes in the population = 4
• n = number of draws = 3
• k = number of successes in the sample = 1

Calculating the probability:

P(X=1) = [C(4,1) * C(14,2)] / C(18,3)

Using combinations:

• C(4,1) = 4
• C(14,2) = (14*13)/2 = 91
• C(18,3) = (181716)/6 = 816

P(X=1) = (4 * 91) / 816 ≈ 364 / 816 ≈ 0.4461

Thus, the probability that exactly one of the three audited returns has a charitable deduction of more than $1,000 is approximately 0.4461, rounded to four decimal places.

Part (b): Calculating the Means

The mean (expected value) of a hypergeometric distribution is given by:

μ = n * (K / N)

where:

• n = 3 (number of sampled returns)
• K = 4 (number of success returns with large charitable contributions)
• N = 18 (total number of returns)

Calculating the population mean (expected number of successes in the sample):

Mean of the distribution of sample means is:

μ = 3 (4 / 18) ≈ 3 0.2222 ≈ 0.6667

The population mean number of successes (returns with >$1,000 contributions) in the entire population is:

(E.g., total expected successes) = (K / N) * total population, which is 4 out of 18, or approximately 0.2222 per return. However, since the question appears to seek the mean of the distribution of sample proportions, the population mean proportion is 4/18 ≈ 0.2222.

Age at First Marriage - Normal Distribution Calculation

The problem states that the average age at which men in the US marry for the first time is 24.8 years with a standard deviation of 2.9 years, and a sample of 58 men is selected. The goal is to determine the probability that this sample's average age is less than 25.2 years. This involves the sampling distribution of the sample mean for a normally distributed population.

First, the standard error (SE) of the mean is calculated as:

SE = σ / √n = 2.9 / √58 ≈ 2.9 / 7.6158 ≈ 0.381

Next, calculate the z-score for the sample mean of 25.2:

z = (X̄ - μ) / SE = (25.2 - 24.8) / 0.381 ≈ 0.4 / 0.381 ≈ 1.05

Using standard normal distribution tables or a calculator, the probability that Z

Therefore, the probability that the average age at first marriage among the sampled men is less than 25.2 years is about 0.8531, rounded to four decimal places.

References

• DeGroot, M. H., & Schervish, J. (2012). Probability and Statistics (4th ed.). Pearson.
• Freudenrich, C. (2020). Hypergeometric Distribution. Encyclopaedia Britannica. https://www.britannica.com/science/hypergeometric-distribution
• Kleinbaum, D. G., Kupper, L. L., Nizam, A., & Rosenberg, E. S. (2013). Applied Regression Analysis and Other Multivariable Methods. Cengage Learning.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers and Scientists (9th ed.). Pearson.
• Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
• Mendenhall, W., Beaver, R., & Beaver, B. (2013). Introduction to Probability and Statistics (14th ed.). Cengage Learning.
• Ross, S. (2009). A First Course in Probability (9th ed.). Pearson.
• Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
• Hogg, R. V., Tanis, E. A., & Zimmerman, D. L. (2014). Probability and Statistical Inference (9th ed.). Pearson.