Question

2. During the 2009 tax filing season, 15.8% of all individual U.S. tax returns were prepared by H&R Block. Suppose we randomly select 3 tax returns.

(a) Describe the probability distribution for X = the number in the sample whose returns were prepared by H&R Block. In other words, for each value of x, determine the associated probability.

(b) What is the mean and standard deviation, respectively, of X?

A. 2.526; 0.399 B. 2.526; 0.632 C. 0.474; 0.399 D. 0.474; 0.632

(c) For the probability distribution modeled in this question, is the assumption that we’re sampling with or without replacement? Explain.

(d) Suppose we wanted to use the normal approximation to the binomial distribution. What are the required conditions to use this approximation and are those conditions met here? Explain.

Answer 1

Let p = probability of the  individual U.S. tax returns were prepared by H&R Block = 0.158, which is constant for each trial

X = the number in the sample whose returns were prepared by H&R Block

n = sample size = 3 , which is fixed.

The outcome of one trial does not affect the outcome of the other trial.

Also there are only two possible outcomes.

So that X follows binomial distribution with parameters n = 3, and p =0.158

The general formula of binomial distribution is as follow:

x = 0, 1, ...,n

Let's find probabilities using the above formula

For X = 0

For x = 1

For x = 2

For x = 3

(b) What is the mean and standard deviation, respectively, of X?

Mean = n*p = 3*0.158 = 0.474

standard deviation =

So correct choice is option D. 0.474; 0.632

(c) For the probability distribution modeled in this question, is the assumption that we’re sampling with or without replacement? Explain.

With replacement

Because the population size is very large and we select sample of size only 3 so we can assume the samples are taken with replacements.

d) (d) Suppose we wanted to use the normal approximation to the binomial distribution. What are the required conditions to use this approximation and are those conditions met here? Explain.

n the sample size is less than the 10% less than the population size N

This condition is satisfied.

n*p >= 10 which is not satisfied

also n*p(1-p) >= which is not satisfied.

Therefore here normal approximation to the binomial distribution is not good to find the probabilities.