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# Distribution

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Suppose that we want to estimate the mean score on a nationwide examination in finance, and for this purpose we choose a random sample of exam scores. The sample we choose has a mean of and a standard deviation of . For each of the following sampling scenarios, determine which test statistic is appropriate to use when making inference statements about the population mean.
(In the table, refers to a variable having a standard normal distribution, and refers to a variable having a t distribution.)
sampling scenario

1. The same has size 13, and it is from a normally distributed population with unknown standard deviation.

a. Z
b. T
c. Could use either Z or T
d. Unclear

2. The sample has size 80, and it is from a non-normally distributed population.

a. Z
b. T
c. Could use either Z or T
d. Unclear

3. The sample has size 12, and it is from a normally distributed population with a known standard deviation of 75.

a. Z
b. T
c. Could use either Z or T
d. Unclear

4. The sample has size 17, and it is from a population with a distribution about which we know very little.

a. Z
b. T
c. Could use either Z or T
d. Unclear
5. The sample has size 90, and it is from a non-normally distributed population with a known standard deviation of 75.
a. Z
b. T
c. Could use either Z or T
d. Unclear

##### Solution Summary

Suppose that we want to estimate the mean score on a nationwide examination in finance, and for this purpose we choose a random sample of exam scores. The sample we choose has a mean of and a standard deviation of . For each of the following sampling scenarios, determine which test statistic is appropriate to use when making inference statements about the population mean.
(In the table, refers to a variable having a standard normal distribution, and refers to a variable having a t distribution.)
sampling scenario

1. The same has size 13, and it is from a normally distributed population with unknown standard deviation.

a. Z
b. T
c. Could use either Z or T
d. Unclear

2. The sample has size 80, and it is from a non-normally distributed population.

a. Z
b. T
c. Could use either Z or T
d. Unclear

3. The sample has size 12, and it is from a normally distributed population with a known standard deviation of 75.

a. Z
b. T
c. Could use either Z or T
d. Unclear

4. The sample has size 17, and it is from a population with a distribution about which we know very little.

a. Z
b. T
c. Could use either Z or T
d. Unclear
5. The sample has size 90, and it is from a non-normally distributed population with a known standard deviation of 75.
a. Z
b. T
c. Could use either Z or T
d. Unclear

##### Solution Preview

1. The same has size 13, and it is from a normally distributed population with unknown standard deviation.
<br>
<br>a. Z
<br>b. T
<br>c. Could use either Z or T
<br>d. Unclear
<br>
<br>*sample size is small, but it is normally distributed, we can use T when the standard deviation is unknown.
<br>
<br>2. The sample has size 80, and it is from a non-normally distributed ...

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