Question 1

Suppose the heights of 3-year-old female children are normally distributed with mean $$\mu = 38.1$$ inches and standard deviation $$\sigma = 1.27$$ inches. What is the $$z-$$score for a 3-year-old girl who is 40 inches tall?

Accepted Answer

$$z = \frac{40 - 38.1}{1.27} \approx 1.50$$

Question 2

A pediatrician wants to know the proportion of 3-year-old females who are shorter than 35 inches. Heights are normally distributed with mean $$\mu = 38.1$$ inches and standard deviation $$\sigma = 1.27$$ inches. Which of the following steps is correct to find this proportion?

Accepted Answer

Calculate $$z = \frac{35 - 38.1}{1.27}$$, then use the standard normal table to find $$P(Z \u003c z).$$

Question 3

If $$X is a $$normally distributed random variable with mean $$\mu$$ and standard deviation $$\sigma$$, what is the formula to convert $$X to $$the standard normal variable $$Z$$?

Accepted Answer

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

Question 4

Which of the following best describes the area under the normal curve to the left of $$z$$?

Accepted Answer

It represents the probability that a standard normal variable is less than $$z.$$

Question 5

A standardized test has scores that are normally distributed with mean $$\mu = 500$$ and standard deviation $$\sigma = 100. $$What score corresponds to the 84th percentile?

Accepted Answer

$$X = 500 + (0.99)(100) = 599$$

Question 6

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

Accepted Answer

The total area under the normal curve is always 1.

Question 7

A SAT math section is approximately normally distributed with mean $$\mu = 514$$ and standard deviation $$\sigma = 118. $$What is the probability that a randomly selected student scores above 650?

Accepted Answer

First, calculate $$z = \frac{650 - 514}{118} \approx 1.15$$, then find $$P(Z \u003e 1.15)$$ using the standard normal table.

Question 8

If you want to find the area between $$z_1$$ and $$z_2$$ under the standard normal curve, which of the following is the correct approach?

Accepted Answer

Find $$P(Z \u003c z_2$) - P(Z \u003c z_1$).$$

Question 9

What is the probability of observing any specific value of a continuous random variable, such as a normal random variable?

Accepted Answer

0

Question 10

A researcher wants to find the height that separates the shortest 20% of 3-year-old females from the rest. Heights are normally distributed with mean $$\mu = 38.1$$ inches and standard deviation $$\sigma = 1.27$$ inches. Which of the following steps is correct?

Accepted Answer

Find the $$z-$$score corresponding to the 20th percentile, then use $$X = \mu + z\sigma.$$

Question 11

Which of the following best describes the notation $$z_{\alpha}$$?

Accepted Answer

It is the $$z-$$score such that the area to the right of $$z_{\alpha} is$$ $$\alpha.$$

Question 12

A hospital records the birth weights of newborns, which are normally distributed with mean $$\mu = 3.5 kg $$and standard deviation $$\sigma = 0.5 kg. $$What is the probability that a randomly selected newborn weighs between 3 kg and 4 kg?

Accepted Answer

Calculate $$z_1$$ = \frac{3 - 3.5}{0.5} = -1$$, z_2$ = \frac{4 - 3.5}{0.5} = 1$$, then find $$P(-1 \u003c Z \u003c 1).$$

Question 13

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

Accepted Answer

The mean, median, and mode are all different.

Question 14

A company finds that the time to assemble a product is normally distributed with mean $$\mu = 30$$ minutes and standard deviation $$\sigma = 4$$ minutes. What is the probability that a randomly selected assembly takes more than 38 minutes?

Accepted Answer

Calculate $$z = \frac{38 - 30}{4} = 2$$, then find $$P(Z \u003e 2).$$

Question 15

If the area to the left of $$z is 0.975$$, what is the value of $$z$$?

Accepted Answer

$$z \approx 1.96$$

Question 16

Which of the following best explains why the probability of a continuous random variable taking any exact value is zero?

Accepted Answer

Because the area under a single point on the curve is zero.

Question 17

A university wants to admit students who score in the top 10% on a standardized test. If the test scores are normally distributed with mean $$\mu = 500$$ and standard deviation $$\sigma = 100$$, what is the minimum score required for admission?

Accepted Answer

Find $$z$$ such that $$P(Z \u003e z) = 0.10$$ ($$z \approx 1.28$$), then $$X = 500 + (1.28)(100) = 628$$

Section 7.2 Applications of the Normal Distribution: Area and Value Calculations

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