Question 1

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

Accepted Answer

The area under the curve is always less than 1.

Question 2

Suppose the heights of three-year-old girls are normally distributed with mean $$\mu = 38.72$$ inches and standard deviation $$\sigma = 3.17$$ inches. What is the $$z-$$score for a girl who is 35 inches tall?

Accepted Answer

$$z = \dfrac{35 - 38.72}{3.17} = -1.17$$

Question 3

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 4

A pediatrician wants to know the proportion of three-year-old girls with heights between 35 and 40 inches. Given $$\mu = 38.72$$ and $$\sigma = 3.17$$, what is the correct process to find this probability?

Accepted Answer

Convert 35 and 40 to $$z-$$scores, find the area to the left of each, and subtract the smaller area from the larger.

Question 5

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?

Accepted Answer

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

Question 6

What is the equation of the least-squares regression line for the data in Table 2, as shown in the study materials?

Accepted Answer

$$\hat{y} = -2.1676x + 18.6704$$

Question 7

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

Accepted Answer

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

Question 8

A student scores 120 on an IQ test with $$\mu = 100$$ and $$\sigma = 15. $$How many standard deviations above the mean is this score?

Accepted Answer

$$z = \dfrac{120 - 100}{15} = 1.33$$

Question 9

If the area to the left of $$z = 1.33$$ under the standard normal curve is $$0.9082$$, what is the area to the right of $$z = 1.33$$?

Accepted Answer

$$1 - 0.9082 = 0.0918$$

Question 10

Which of the following best describes the inflection points of a normal curve?

Accepted Answer

They occur at $$x = \mu - \sigma$$ and $$x = \mu + \sigma.$$

Question 11

A SAT math score is normally distributed with $$\mu = 516$$ and $$\sigma = 116. $$What SAT score corresponds to the 95th percentile? (Use $$z = 1.645$$ for the 95th percentile.)

Accepted Answer

$$x = 516 + (1.645)(116) = 707$$

Question 12

Which of the following is the correct interpretation of the area under a normal curve between $$x = a$$ and $$x = b$$?

Accepted Answer

It represents the probability that a randomly selected individual has a value between $$a$$ and $$b.$$

Question 13

If a normal random variable $$X$$ has mean $$\mu$$ and standard deviation $$\sigma$$, what is the formula to convert a value $$x to a$$ $$z-$$score?

Accepted Answer

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

Question 14

A continuous random variable $$X is $$normally distributed. Which of the following statements is TRUE?

Accepted Answer

$$P(X = a) = 0$$ for any specific value $$a.$$

Question 15

The heights of three-year-old girls are normally distributed with $$\mu = 38.72$$ inches and $$\sigma = 3.17$$ inches. What is the height at the 20th percentile? (Use $$z = -0.84$$ for the 20th percentile.)

Accepted Answer

$$x = 38.72 + (-0.84)(3.17) = 36.05$$ inches

Question 16

Which of the following best describes the effect of increasing the standard deviation $$\sigma of a $$normal distribution while keeping the mean $$\mu$$ constant?

Accepted Answer

The curve becomes flatter and more spread out.

Question 17

If the area to the left of $$z = -1.17 is$$ $$0.1210$$ and the area to the left of $$z = 0.40 is$$ $$0.6554$$, what is the probability that $$z is $$between $$-1.17$$ and $$0.40$$?

Accepted Answer

$$0.6554 - 0.1210 = 0.5344$$

Question 18

Which of the following is a correct application of the Empirical Rule?

Accepted Answer

Approximately 99.7% of data falls within three standard deviations of the mean.

Question 19

A normal distribution has $$\mu = 50$$ and $$\sigma = 5. $$What is the probability that a randomly selected value is exactly 50?

Accepted Answer

0

Least-Squares Regression and The Normal Probability Distribution

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