Q: A test has $$\mu=100$$ and $$\sigma=20. If a $$student’s z-score is $$-1.5$$, what is their raw score?

$$X = 100 + 20 \times (-1.5) = 70$$

Question 1

Suppose you received a score of $$X=60 on a $$statistics exam and a score of $$X=56 on a $$research methods test. The exams had the following means and standard deviations: Research Methods: $$\mu=48$$, $$\sigma=4$$; Statistics: $$\mu=50$$, $$\sigma=10. $$For which course should you expect a better grade, based on your z-scores?

Accepted Answer

Statistics, because the z-score is higher.

Question 2

Given a population with mean $$\mu=44$$ and standard deviation $$\sigma=6$$, a new distribution is created with $$\mu=50$$ and $$\sigma=10. $$What is the new standardized value for a score of $$X=47$$ from the original distribution?

Accepted Answer

$$z = \frac{47-44}{6} = 0.5$$; new standardized value: $$X' = 50 + 10 	imes 0.5 = 55$$

Question 3

An individual has a new standardized score of $$X'=65 in $$the new distribution ($$\mu=50$$, $$\sigma=10). $$What was this person’s score in the original distribution ($$\mu=44$$, $$\sigma=6$$)?

Accepted Answer

$$z = \frac{65-50}{10} = 1.5$$; original score: $$X = 44 + 6 	imes 1.5 = 53$$

Question 4

A client has an investment portfolio with $$\mu = \500$$,000$$ and $$\sigma = \$$15,000. $$What is the probability that the value of her portfolio is between $$\485$$,000$$ and $$\$$530,000$$?

Accepted Answer

Find $$z-$$scores: $$z_1$$ = \frac{485,000 - 500,000}{15,000} = -1$$, z_2$ = \frac{530,000 - 500,000}{15,000} = 2. $$Probability is $$P(-1 \u003c z \u003c 2) = 0.9772 - 0.1587 = 0.8185$$

Question 5

Which formula is used to calculate a z-score for a value $$X in a $$distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 6

If a distribution is standardized, what are the mean and standard deviation of the new distribution?

Accepted Answer

Mean $$= 0$$, Standard deviation $$= 1$$

Question 7

A student scores $$X=70 on a $$test with $$\mu=60$$ and $$\sigma=5. $$What is the student’s z-score?

Accepted Answer

$$z = \frac{70-60}{5} = 2$$

Question 8

If a z-score is negative, what does this indicate about the value of $$X$$ compared to the mean?

Accepted Answer

$$X is $$below the mean.

Question 9

A researcher wants to compare scores from two different tests with different means and standard deviations. What statistical method should be used?

Accepted Answer

Standardize the scores using z-scores.

Question 10

If the probability that a portfolio value is less than $$\485$$,000$$ is 0.1587$, and the probability that it is less than $$\$$530,000 is$$ $$0.9772$$, what is the probability that the value is between $$\485$$,000$$ and $$\$$530,000$$?

Accepted Answer

$$0.9772 - 0.1587 = 0.8185$$

Question 11

A test has $$\mu=100$$ and $$\sigma=20. If a $$student’s z-score is $$-1.5$$, what is their raw score?

Accepted Answer

$$X = 100 + 20 	imes (-1.5) = 70$$

Question 12

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

Approximately 68% of values lie within one standard deviation of the mean.

Question 13

A portfolio value is normally distributed with $$\mu=500,000$$ and $$\sigma=15,000. $$What is the z-score for a value of $$530,000$$?

Accepted Answer

$$z = \frac{530,000 - 500,000}{15,000} = 2$$

Question 14

If a score is one standard deviation above the mean, what is its z-score?

Accepted Answer

$$z = 1$$

Question 15

A population has $$\mu=44$$ and $$\sigma=6. $$What is the z-score for $$X=38$$?

Accepted Answer

$$z = \frac{38-44}{6} = -1$$

Question 16

Why is standardizing scores useful when comparing results from different distributions?

Accepted Answer

It allows comparison by removing the effects of different means and standard deviations.

Question 17

If a test score is at the mean, what is its z-score?

Accepted Answer

$$z = 0$$

Question 18

A student’s z-score on a test is $$2.5. $$What does this indicate about their performance?

Accepted Answer

The student scored 2.5 standard deviations above the mean.

Standardization, Z-Scores, and the Normal Distribution

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