# Algebra II : Z-scores

## Example Questions

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### Example Question #1 : Z Scores

A large group of test scores is normally distributed with mean 78.2 and standard deviation 4.3. What percent of the students scored 85 or better (nearest whole percent)?

Explanation:

If the mean of a normally distributed set of scores is  and the standard deviation is , then the -score corresponding to a test score of  is

From a -score table, in a normal distribution,

We want the percent of students whose test score is 85 or better, so we want . This is

or about 5.7 % The correct choice is 6%.

### Example Question #2 : Z Scores

The salaries of employees at XYZ Corporation follow a normal distribution with mean 60,000 and standard deviation 7,500. What proportion of employees earn approximately between 69,000 and 78,000?

Use the normal distribution table to calculate the probabilities. Round your answer to the nearest thousandth.

Explanation:

Let X represent the salaries of employees at XYZ Corporation.

We want to determine the probability that X is between 69,000 and 78,000:

To approximate this probability, we convert 69,000 and 78,000 to standardized values (z-scores).

We then want to determine the probability that z is between 1.2 and 2.4

The proportion of employees who earn between 69,000 and 78,000 is 0.107.

### Example Question #1 : Z Scores

On a statistics exam, the mean score was and there was a standard deviation of . If a student's actual score of , what is his/her z-score?

Explanation:

The z-score is a measure of an actual score's distance from the mean in terms of the standard deviation. The formula is:

Where  are the mean and standard deviation, respectively. is the actual score.

If we plug in the values we have from the original problem we have

which is approximately .

### Example Question #4 : Z Scores

A distributor manufactures a product that has an average weight of  pounds.

If the standard deviation is  pounds, determine the z-score of a product that has a weight of  pounds.

Explanation:

The z-score can be expressed as

where

Therefore the z-score is:

### Example Question #5 : Z Scores

The mean grade on a science test was 79 and there was a standard deviation of 6. If your sister scored an 88, what is her z-score?

Explanation:

Use the formula for z-score:

Where  is her test score,  is the mean, and  is the standard deviation.

### Example Question #6 : Z Scores

You just took your ACT. The mean score was a 22 with a standard deviation of 3. If you scored a 26, what is your z-score?

Explanation:

Use the formula for z-score:

Where  is your score,  is the mean, and  is the standard deviation.

### Example Question #7 : Z Scores

Your teacher tells you that the mean score for a test was a  and that the standard deviation was  for your class.

You are given that the -score for your test was  . What did you score on your test?

Explanation:

The formula for a z-score is

Plugging in your z-score, mean, and standard deviation that was originally given in the question we get the following.

Now to find the grade you got on the test we will solve for .

### Example Question #2 : Z Scores

In a normal distribution, if the mean score is 8 in a gymnastics competition and the student scores a 9.3, what is the z-score if the standard deviation is 2.5?

Explanation:

Write the formula to find the z-score.  Z-scores are defined as the number of standard deviations from the given mean.

Substitute the values into the formula and solve for the z-score.

### Example Question #3 : Z Scores

Suppose a student scored a  on a test.  The mean of the tests are , and the standard deviation is .  What is the student's z-score?

Explanation:

Write the formula for z-score where  is the data,  is the population mean, and  is the population standard deviation.

Substitute the variables.

The z-score is:

### Example Question #10 : Z Scores

Suppose Bob's test score is 50.  Determine the z-score if the standard deviation is 3, and the mean is 75.

Explanation:

Write the formula for z-scores.  This tells how many standard deviations above the below the mean.

Substitute the known values into the equation.

The z-score is:

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