Question 1

Which of the following z-scores is closest to the mean?

Accepted Answer

The mean has a z-score of 0, and since -.10 is the closest to 0, it is the best choice.

Question 2

Which of the following z-scores is furthest away from the mean?

Accepted Answer

A z-score of +2.00 is further away from the mean than any negative z-score or a z-score of 0.

Question 3

A set of scores has a standard deviation of 6. A raw score of 53 converts to a z-score of -1.5. What is the mean of the distribution?

Accepted Answer

The formula for calculating a z-score is $z = \frac{(X - \mu)}{\sigma}$, where $X$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the given values, $-1.5 = \frac{(53 - \mu)}{6}$. Solving for $\mu$ gives $\mu = 62$.

Question 4

A set of scores has a σ = 4. Susan's score was 93 which converted to a z-score of +2.00. What was the mean of the distribution?

Accepted Answer

Z-score formula: z = (x - μ) / σ

Given that Susan's z-score is 2.00 and the population standard deviation (σ) is 4, we can find her score (x):

2.00 = (93 - μ) / 4

8 = 93 - μ

μ = 93 - 8 = 85

Therefore, the mean of the distribution is 85.

Question 5

Hans scored 16 points above the mean with a z-score of +2.00. What was the standard deviation for the distribution?

Accepted Answer

The z-score formula is Z = (X - μ) / σ, where Z is the z-score, X is the value, μ is the mean, and σ is the standard deviation. Given Z = +2.00 and X - μ = 16, solving for σ gives σ = (X - μ) / Z = 16 / 2 = 8.

Question 6

A set of scores has a mean of 75. Jiang's score of 60 converted to a z-score of -1.5. What was the standard deviation of the distribution?

Accepted Answer

We can use the formula for converting a score to a z-score:

z = (X - µ) / σ

where z is the z-score, X is the score, µ is the mean, and σ is the standard deviation.

We know that Jiang's score of 60 has a z-score of -1.5:

-1.5 = (60 - 75) / σ

Solving for σ, we get:

σ = (60 - 75) / -1.5 = 10

Therefore, the standard deviation of the distribution is 10, which is answer choice B.

Question 7

A set of scores has a mean of 36 and a standard deviation of 6. Karim's z-score was -1.50. What was his raw score?

Accepted Answer

We use the formula z = (X - μ) / σ to find the raw score. Plugging in the given values, we have -1.50 = (X - 36) / 6. Solving for X, we get X = 27.

Question 8

Julie's z-score was +2.50 in a distribution whose μ = 100 with a σ = 10. What was Julie's raw score?

Accepted Answer

The formula for finding a raw score from a z-score is: 
raw score = (z-score x standard deviation) + mean. 
Substitute the given values into the formula: 
raw score = (2.50 x 10) + 100 
raw score = 25 + 100 
raw score = 125

Question 9

A set of scores has a μ = 65 and a σ = 12. Lucinda's z-score was +1.50. What was her raw score?

Accepted Answer

To find Lucinda's raw score, we can use the following formula:

z = (x - μ) / σ

where z is the z-score, x is the raw score, μ is the mean, and σ is the standard deviation.

Substituting the given values, we get:

1.50 = (x - 65) / 12

Multiplying both sides by 12, we get:

18 = x - 65

Adding 65 to both sides, we get:

x = 83

Therefore, Lucinda's raw score was 83.

Question 10

Elijah's z-score was -.25 in a distribution where the mean was 70 and the standard deviation was 8. What is his raw score?

Accepted Answer

The answer of Elijah's z-score was -.25 in a distribution...

Question 11

The WAIS has a mean of 100 and a standard deviation of 15. One of the requirements to become a member of MENSA, an organization for smart people, is to score a minimum of two standard deviations above the mean on this test. What is the minimum score requirement?

Accepted Answer

Two standard deviations above the mean would be 2 x 15 = 30 points. Adding this to the mean of 100 gives a minimum score requirement of 130. Therefore, the correct answer is B.

Question 12

Which of the following statistics would be helpful for comparing the scores from two different distributions?

Accepted Answer

The z-score is useful for comparing scores from different distributions because it standardizes scores by converting them into a common scale based on the mean and standard deviation of the distribution.

Question 13

The mean for a statistics exam was 85. Karen calculated her own z-score to be +1.75 for her test result of 68 points. What is wrong with Karen's calculation?

Accepted Answer

Karen's z-score should be negative because her test result (68) is below the mean (85).

Question 14

Jorge scored 70 on an exam and his z-score was -2.00. Michelle scored 82 on the same test and her z-score was +1.00. What was the mean of the exam?

Accepted Answer

We know that the z-score formula is:

z = (x - mean) / standard deviation

We can rearrange this formula to solve for the mean:

mean = x - (z * standard deviation)

For Jorge, we know that his z-score is -2.00 and his score was 70. Therefore:

70 = mean - (2 * standard deviation)

For Michelle, we know that her z-score is +1.00 and her score was 82. Therefore:

82 = mean + (1 * standard deviation)

Now we have two equations with two unknowns (mean and standard deviation). We can solve for one of the unknowns and then substitute into the other equation to solve for the other unknown.

Let's solve for standard deviation first. We can rearrange the first equation as:

standard deviation = (mean - 70) / 2

Substituting this into the second equation, we get:

82 = mean + ((mean - 70) / 2)

Simplifying:

164 = 3 * mean - 70

3 * mean = 234

mean = 78

Therefore, the answer is B) 78.

Question 15

Isabelle scored 98 on an exam and her z-score was +3.00. Isaac scored 80 on the same exam and his z-score was -3.00. What was the mean of the exam?

Accepted Answer

The mean of the exam can be found by averaging the scores of Isabelle and Isaac adjusted for their z-scores. Since Isabelle's z-score is +3, it means her score is 3 standard deviations above the mean. Similarly, Isaac's z-score of -3 means his score is 3 standard deviations below the mean. The difference between their scores is 18 (98 - 80), which equals 6 standard deviations (3 for Isabelle and 3 for Isaac). Therefore, one standard deviation equals 3 points (18/6). The mean is then found by either adding 3 standard deviations to Isaac's score or subtracting 3 standard deviations from Isabelle's score. Adding 3 standard deviations (9 points) to Isaac's score of 80 gives us 89, or subtracting 9 points from Isabelle's score of 98 also gives us 89.

Question 16

If one z-score from a distribution is -1 for a raw score of 66 and another z-score is +3 for a raw score of 78, what is the standard deviation for the distribution?

Accepted Answer

The standard deviation is 3 because the difference between the two raw scores is 12 (78 - 66), and the difference between the two z-scores is 4 (3 - (-1)). Therefore, each standard deviation unit is equal to 3 raw score points (12 / 4 = 3).

Question 17

Ariana's raw score of 33 on her philosophy exam converted to a z-score of -1.50. Claire's raw score on the same exam was 48 and it converted to a z-score of +1.00. What was the standard deviation for the distribution?

Accepted Answer

We can use the formula for z-score:

z = (x - mean) / standard deviation

For Ariana: 
-1.50 = (33 - mean) / standard deviation

For Claire: 
1.00 = (48 - mean) / standard deviation

We can solve for the mean using either equation:

mean = x - z * standard deviation

For Ariana: 
mean = 33 - (-1.50) * standard deviation = 33 + 1.50 * standard deviation

For Claire: 
mean = 48 - (1.00) * standard deviation = 48 - 1.00 * standard deviation

Setting these two expressions equal to each other:

33 + 1.50 * standard deviation = 48 - 1.00 * standard deviation

Solving for standard deviation:

2.50 * standard deviation = 15

standard deviation = 6

Therefore, the answer is B) 6.

Question 18

You scored at the mean on your statistics exam. What is your z-score?

Accepted Answer

Since the student scored at the mean, their score is equal to the population mean. Therefore, their deviation from the mean (or z-score) is 0.

Question 19

To change a raw score to a z-score, you need to

Accepted Answer

To convert a raw score to a z-score, you need to subtract the mean of the distribution from the raw score and then divide that result by the standard deviation of the distribution. Therefore, you need to know the mean and standard deviation of the distribution.

Question 20

To use the TSS formula,

Accepted Answer

The TSS formula requires raw scores to be converted to z-scores, and both the mean and standard deviation to be specified.

Quiz 5: Z-Scores and Other Standard Scores