Algebra II : Z-scores

Study concepts, example questions & explanations for Algebra II

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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)?

Possible Answers:

Correct answer:

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?

 

Normal-distribution

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

Possible Answers:

Correct answer:

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 #3 : Graphing Data

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?

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

Explanation:

Use the formula for z-score:

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

Example Question #6 : 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?

Possible Answers:

Correct answer:

Explanation:

The formula for a z-score is

where   = mean and  = standard deviation and =your test grade.

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 #6 : 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?

Possible Answers:

Correct answer:

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 #2 : 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?

Possible Answers:

Correct answer:

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 #8 : Z Scores

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

Possible Answers:

Correct answer:

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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