According to an IRS study, it takes an average of 330 minutes for taxpayers to prepare, copy, and electronically file a 1040 tax form. A consumer watchdog agency selects a random sample of 40 taxpayers and finds the standard deviation of the time to prepare, copy, and electronically file for 1040 is 80 minutes.

a) What assumption or assumptions do you need to make about the shape of the Population?

b) What is the standard error of the mean in this example?

c) What is the likelihood the sample mean is greater than 320?

d) What is the likelihood the sample mean is between 320 and 350 minutes?

e) What is the likelihood the sample mean is greater than 350 minutes?

**Solution: **(a) We usually assume that the population has a normal distribution, which means that the distribution is bell-shaped, and symmetric with respect to the mean. We also assume independent sampling.

(b) The standard error of the mean is defined as

_{}

where_{}is the sample standard deviation, and_{}is the sample size. In our case_{}and_{}. Therefore, the standard error of the mean is equal to

_{}

(c) Let_{} be the random preparation time. We need to compute the following probability:

_{}

We need to compute the z-score of 320, which is

_{}

This way, the probability we need to compute is equal to

_{}

In order to calculate this probability, we use the EXCEL function NORMSDIST, which computes the cumulative distribution for a standard normal distribution. We know that

_{}

Therefore,

_{}

(d) We need now to compute the following probability:

_{}

We need to compute the z-score of 320 and 350. They are computed as

_{}

which is the z-score of 320, and

_{}

which is the z-score of 350. This way, the probability we need to compute is equal to

_{}

In order to calculate this probability, we use the EXCEL function NORMSDIST, which leads us to

_{}

(e) Finally, we need to compute the following probability:

_{}

We need to compute the z-score of 350. It is computed as

_{}

This way, the probability we need to compute is equal to

_{}

In order to calculate this probability, we use the EXCEL function NORMSDIST, which leads us to

_{}

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## Stats Solution Summary

One of the most useful tools to work with normal probability distributions is Excel, which has functions that allow to work with standard normal distribution and general normal distributions.