Question 1

Which of the following best describes the normal distribution?

Accepted Answer

It is a symmetric, bell-shaped distribution characterized by its mean $$\mu$$ and standard deviation $$\sigma.$$

Question 2

According to the empirical rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

$$95\%$$

Question 3

If the heights of adult women are normally distributed with $$\mu = 65$$ inches and $$\sigma = 3.5$$ inches, what is the interval that contains almost all (99.7%) of the heights?

Accepted Answer

$$[54.4, 75.5]$$

Question 4

What is the formula for calculating the z-score for a value $$x in a $$normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 5

A student scores 650 on the SAT ($$\mu = 500$$, $$\sigma = 100) $$and another scores 30 on the ACT ($$\mu = 21$$, $$\sigma = 4.7). $$Who performed better relative to their test's distribution?

Accepted Answer

The ACT student, because their z-score is higher.

Question 6

What is the probability that a value falls more than 1.43 standard deviations from the mean in a normal distribution?

Accepted Answer

$$0.1528$$

Question 7

Which statement about the standard normal distribution is correct?

Accepted Answer

It has a mean of $$\mu = 0$$ and a standard deviation of $$\sigma = 1$$

Question 8

If a cumulative probability of $$0.025 is $$found in the body of the standard normal table, what is the corresponding z-score?

Accepted Answer

$$z = -1.96$$

Question 9

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is always discrete.

Question 10

A researcher wants to find the probability that a value falls below $$z = 1.43 in a $$standard normal distribution. What is the correct approach?

Accepted Answer

Look up $$z = 1.43 in $$the standard normal table to find the cumulative probability.

Question 11

If $$x is a $$value from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, and its z-score is $$2$$, what is $$x$$?

Accepted Answer

$$x = \mu + 2\sigma$$

Question 12

Why is the empirical rule stated as approximate rather than exact?

Accepted Answer

Because it refers to approximately bell-shaped distributions, not just the normal distribution.

Question 13

Which of the following real-world scenarios is most likely to be modeled by a normal distribution?

Accepted Answer

The heights of adult men in a population.

Question 14

If the mean score on a test is $$80$$ and the standard deviation is $$5$$, what is the z-score for a student who scored $$90$$?

Accepted Answer

$$z = 2$$

Question 15

Which step is NOT required when using Table A to find the cumulative probability for a z-score?

Accepted Answer

Multiply the z-score by the mean.

Question 16

A value $$x is 1.5 $$standard deviations above the mean in a normal distribution. What is its z-score?

Accepted Answer

$$z = 1.5$$

Question 17

Which of the following statements is true about z-scores?

Accepted Answer

They measure the number of standard deviations a value is from the mean.

Question 18

If a normal random variable has mean $$\mu$$ and standard deviation $$\sigma$$, what is the probability of falling within one standard deviation of the mean?

Accepted Answer

$$0.68$$

Question 19

Which of the following is a correct application of the 68-95-99.7 rule?

Accepted Answer

About 68% of values fall within $$\mu \pm \sigma.$$

(Lecture 14) Normal Distributions and the 68-95-99.7 Rule

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