AP Statistics - Data Sets And Z-scores

Your professor gave back the mean and standard deviation of your class's scores on the last exam.

$\mu=75$

$\sigma=5$

Your friend says the z-score of her exam is .

What did she score on her exam?

Explanation

The z-score is the number of standard deviations above the mean.

We can use the following equation and solve for x.

$\frac{x-\mu}{\sigma}=z$

$\frac{x-75}{5}=2$

Two standard deviations above 75 is 85.

All of the students at a high school are given an entrance exam at the beginning of 9th grade. The scores on the exam have a mean of and a standard deviation of . Sally's z-score is . What is her score on the test?

Explanation

The z-score equation is $z= (x-\mu)/\sigma$ .

To solve for we have .

Your boss gave back the mean and standard deviation of your team's sales over the last month.

$\mu=35$

$\sigma=3$

Your friend says the z-score of her number of sales is .

How many sales did she make?

Explanation

The z-score is the number of standard deviations above or below the mean.

We can use the known information with the following formula to solve for x.

$\frac{x-\mu}{\sigma}=z$

$\frac{x-35}{3}=-1$

Your teacher gives you the z-score of your recent test, and says that the mean score was a 60, with a standard deviation of 6. Your z-score was a -2.5. What did you score on the test?

Explanation

To find out your score on the test, we enter the given information into the z-score formula and solve for .

$z=\frac{x-\mu }{\sigma }$ where is the z-score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

As such,

$-2.5=\frac{x- 60}{6 }$

So you scored a on the test.

Your professor gave back the mean and standard deviation of your class's scores on the last exam.

$\mu=75$

$\sigma=5$

Your friend says the z-score of her exam is .

What did she score on her exam?

Explanation

The z-score is the number of standard deviations above the mean.

We can use the following equation and solve for x.

$\frac{x-\mu}{\sigma}=z$

$\frac{x-75}{5}=2$

Two standard deviations above 75 is 85.

Explanation

The z-score equation is $z= (x-\mu)/\sigma$ .

To solve for we have .

Your boss gave back the mean and standard deviation of your team's sales over the last month.

$\mu=35$

$\sigma=3$

Your friend says the z-score of her number of sales is .

How many sales did she make?

Explanation

The z-score is the number of standard deviations above or below the mean.

We can use the known information with the following formula to solve for x.

$\frac{x-\mu}{\sigma}=z$

$\frac{x-35}{3}=-1$

Your teacher gives you the z-score of your recent test, and says that the mean score was a 60, with a standard deviation of 6. Your z-score was a -2.5. What did you score on the test?

Explanation

To find out your score on the test, we enter the given information into the z-score formula and solve for .

$z=\frac{x-\mu }{\sigma }$ where is the z-score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

As such,

$-2.5=\frac{x- 60}{6 }$

So you scored a on the test.

This past week, the temperature has fluctuated quite a bit. From Monday to Sunday, temperatures in Fahrenheit have been the following: 62, 68, 52, 40, 78, 72, 60. The standard deviation is given as 13 and does not need to be calculated. Convert these daily temperatures into z-scores and give the days that are at least one standard deviation away from the mean.

Thursday and Friday

Thursday

Wednesday

Thursday and Wednesday

Friday

Explanation

To convert the temperatures into the z-scores, subtract the mean temperature (61.7) from the daily temperatures and divide this by the given standard deviation (13). The daily z-scores are .02, .53, -.81, -1.8, 1.4, .86, and -.14. Recall that z-scores tell you how many standard deviations away from the mean a given observation is.

Only Thursday and Friday are greater than one.

The following data set represents Mr. Marigold's students' scores on the final. The standard deviation for this data set is 8.41. How many standard deviations are you away from the mean if you scored an 86? \[find your z-score\]

$\small \small \small 66, 69, 71, 71, 72, 73.5, 74, 74, 76.5, 76.5, 77.5, 78, 81$
$\small 81, 85, 86, 86, 86, 87, 87.5, 88.5, 89, 90, 92, 92, 100$