Question 1

Suppose the random variable $$X$$ represents the time (in minutes) your friend is late, uniformly distributed between 0 and 30 minutes. What is the probability that your friend is between 10 and 20 minutes late?

Accepted Answer

$$P(10 \leq X \leq 20) = \frac{20-10}{30-0} = \frac{10}{30} = 0.333$$

Question 2

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

Accepted Answer

The curve is skewed to the right.

Question 3

Given a normal distribution with 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 4

The probability that a continuous random variable $$X$$ takes any exact value (e.g., $$X = 12.5) is$$:

Accepted Answer

$$P(X = a) = 0$$

Question 5

A pediatrician finds that the heights of three-year-old females are approximately normally distributed with $$\mu = 38.72$$ inches and $$\sigma = 3.17$$ inches. What proportion of three-year-old females are shorter than 35 inches? (Use $$z = \frac{35 - 38.72}{3.17})$$

Accepted Answer

Find $$z = \frac{35 - 38.72}{3.17} = -1.17$$, so the area to the left is approximately $$0.1210$$

Question 6

If the area under the normal curve to the right of $$x = 200 is$$ $$0.2903$$, what does this mean in the context of cholesterol levels for males aged 20-29 (mean $$\mu = 180$$, $$\sigma = 36.2$$)?

Accepted Answer

About 29.03% of males aged 20-29 have cholesterol levels above 200 mg/dL.

Question 7

Which histogram below best represents a distribution that could be modeled by a normal distribution?

Accepted Answer

A histogram that is bell-shaped, symmetric, and has a single peak.

Question 8

Given a normal distribution with $$\mu = 516$$ and $$\sigma = 116$$, what is the probability that a randomly selected SAT math score is more than 600? (Find $$z$$ and use the standard normal table.)

Accepted Answer

First, $$z = \frac{600 - 516}{116} = 0.724. $$Area to the left is about $$0.765. So$$, $$P(X \u003e 600) = 1 - 0.765 = 0.235.$$

Question 9

If the area to the left of $$x = 259$$ days in a normal distribution of pregnancy lengths ($$\mu = 266$$, $$\sigma = 16) is$$ $$0.3309$$, what does this mean?

Accepted Answer

About 33.09% of pregnancies last 259 days or less.

Question 10

Which of the following best describes how to find the area between $$x_1$$ and $$x_2$$ under a normal curve?

Accepted Answer

Convert $$x_1$$ and $$x_2 to$$ $$z-$$scores, find the area to the left of each, and subtract: $$P(x_1$ \u003c X \u003c x_2$) = P(Z \u003c z_2$) - P(Z \u003c z_1$).$$

Question 11

A normal probability plot is used to:

Accepted Answer

Assess whether data are approximately normally distributed by checking if points form a straight line.

Question 12

If a random variable $$X is $$uniformly distributed on $$[0, 10]$$, what is the probability that $$X = 4.35$$?

Accepted Answer

$$P(X = 4.35) = 0$$

Question 13

Given a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, what is the formula to find the value $$x$$ corresponding to a given $$z-$$score?

Accepted Answer

$$x = \mu + z\sigma$$

Question 14

Which of the following is a key feature of a frequency histogram that suggests the data is approximately normal?

Accepted Answer

The histogram is symmetric and bell-shaped with few outliers.

Question 15

Suppose the area to the left of $$z = -1.28 is$$ $$0.1003. $$What percentile does this $$z-$$score represent?

Accepted Answer

The 10th percentile.

Question 16

If a histogram of restaurant wait times is roughly symmetric and bell-shaped, what can you conclude?

Accepted Answer

The wait times can likely be modeled by a normal distribution.

Question 17

Which of the following statements about the area under a normal curve is true?

Accepted Answer

The total area under the curve is always 1.

Chapter 6: The Normal Probability Distribution – Study Notes

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