Question

According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form $1040$is $10.53$hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample $36$taxpayers.

a. In words, ${\rm X}=\_\_\_\_\_\_\_\_\_\_\_\_\_$

b. In words,$\overline{X}=\_\_\_\_\_\_\_\_\_\_\_\_\_$

c. $X¯~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. Would you be surprised if the $36$taxpayers finished their Form $1040$s in an average of more than $12$hours? Explain why or why not in complete sentences.

e. Would you be surprised if one taxpayer finished his or her Form $1040$in more than $12$hours? In a complete sentence, explain why.

Short answer

a. length of time for an individual to complete IRS

b. mean length of time for a sample of $36$taxpayers to complete the IRS form

c. $X~N\left(10.53,\frac{1}{3}\right)$

d. Yes. I would be surprised because the probability is almost$0$

e. No. I would not be totally surprised because the probability is$0.2312$

Step-by-step solution

Given information

the average length of time for an individual to complete (keep records for,

learn, prepare, copy, assemble, and send) IRS Form$1040$ is $10.53$hours (without any attached schedules).

The distribution is unknown.

Let us assume that the standard deviation is two hours.

Explanation (part a)

The definition of random variable is given by,

$X=$length of time for an individual to complete IRS form $1040$, in hours.

Explanation (part b)

In other words, the sample random variable is defined as, $\overline{X}=$the mean length of time for a sample of $36$taxpayers to complete IRS form.

Explanation (part c)

The random variable can be expressed as,$X~N\left(10.53,\frac{1}{3}\right)$normal distribution.

Explanation (part d)

On taking normal CDF, we get the probability is$0$

Yes. I would be surprised because the probability is almost$0$

Explanation (part e)

If one taxpayer finished his or her Form in more than twelve hours, the probability is about$0.2312$

No. I would not be totally surprised because the probability is $0.2312$