Question 1

Which of the following best describes what a z-score represents in statistics?

Accepted Answer

The difference between a value and the mean, measured in units of standard deviation.

Question 2

Given a sample with mean $$\bar{x} = 20$$ and sample standard deviation $$s = 4$$, what is the z-score for a value $$x = 28$$?

Accepted Answer

$$z = \frac{28 - 20}{4} = 2$$

Question 3

If a value in a population has a z-score of $$-2.5$$, what does this indicate about the value?

Accepted Answer

It is 2.5 standard deviations below the mean.

Question 4

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the z-score for $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 5

Which formula would you use to calculate the z-score for a value from a sample?

Accepted Answer

$$z = \frac{x - \bar{x}}{s}$$

Question 6

A student scores 85 on a test where the mean is 75 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 75}{5} = 2$$

Question 7

If a newborn baby weighs 3 pounds above the average, and the standard deviation for newborn weights is 1 pound, what is the z-score for this baby?

Accepted Answer

$$z = \frac{3}{1} = 3$$

Question 8

Which of the following statements is TRUE regarding z-scores in a bell-shaped (normal) distribution?

Accepted Answer

Almost all values have z-scores between $$-3$$ and $$+3.$$

Question 9

A sample has $$\bar{x} = 50$$, $$s = 8. $$What is the z-score for $$x = 42$$?

Accepted Answer

$$z = \frac{42 - 50}{8} = -1$$

Question 10

If a value has a z-score of $$0$$, what does this mean?

Accepted Answer

The value is equal to the mean.

Question 11

A test has a mean of $$60$$ and a standard deviation of $$12. If a $$student scores $$84$$, what is their z-score?

Accepted Answer

$$z = \frac{84 - 60}{12} = 2$$

Question 12

Why is it important to compare deviations to the standard deviation when interpreting z-scores?

Accepted Answer

Because the standard deviation provides a scale to judge whether a deviation is large or small relative to the data set.

Question 13

Chebyshev's theorem states that at least what percentage of values in any data set have z-scores between $$-3$$ and $$+3$$?

Accepted Answer

At least 89%

Question 14

A value in a sample is $$x = 15$$, the sample mean is $$\bar{x} = 10$$, and the sample standard deviation is $$s = 2. $$What is the z-score?

Accepted Answer

$$z = \frac{15 - 10}{2} = 2.5$$

Question 15

If a value in a data set has a z-score of $$4$$, what can you conclude?

Accepted Answer

The value is unusually far from the mean, more than 3 standard deviations above the mean.

Question 16

A sample has values $$x = 8, 9, 10, 13$$ with mean $$\bar{x} = 10$$ and sample standard deviation $$s = 2.2. $$What is the z-score for $$x = 8$$?

Accepted Answer

$$z = \frac{8 - 10}{2.2} \approx -0.91$$

Question 17

In a population, the mean is $$\mu = 150$$ and the standard deviation is $$\sigma = 10. $$What is the z-score for a test score of $$145$$?

Accepted Answer

$$z = \frac{145 - 150}{10} = -0.5$$

Question 18

Which of the following is NOT a correct interpretation of a z-score?

Accepted Answer

A z-score tells you the probability of a value occurring.

Question 19

A value in a population is $$x = 120$$, the mean is $$\mu = 150$$, and the standard deviation is $$\sigma = 10. $$What is the z-score?

Accepted Answer

$$z = \frac{120 - 150}{10} = -3$$

Understanding Z-Scores in Descriptive Statistics

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