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?

(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

(a)

Binomial Distribution

n = 3

p = 0.158

q = 1 - p = 0.842

So,

Probability Distribution of X is got as follows:

x	p
0	0.5970
1	0.3360
2	0.0631
3	0.0039
Total

(b)

x	p	p x	p   x2
0	0.5970	0	0
1	0.3360	0.3360	0.3360
2	0.0631	0.1262	0.2524
3	0.0039	0.0117	0.0351
Total

So,

Standard Deviation =

(c)

For the probability distribution modeled in this question, the assumption is that we’re sampling with replacement because in that case only the probability of success = p = 0.158 will be constant in each trial.

(d)

Conditions to use the normal approximation to the binomial distribution :

(i) np = 3 X 0.158 = 0.474 is not greater than 5

(ii) nq = 3 X 0.842 = 2.526 is not greater than 5

It is noted that both np and nq are not greater than 5. So, Conditions to use the normal approximation to the binomial distribution are not met.