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

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

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

The total area under the normal curve is always 1.

Q: 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?

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

Q: 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?

Find $P(Z < z_2) - P(Z < z_1)$.

Q: 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?

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

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

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

Q: 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?

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

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

The mean, median, and mode are all different.

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 $P(Z < z)$.

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 $P(Z > 1.15)$ 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 $P(Z < z_2) - P(Z < z_1)$.

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 $P(-1 < Z < 1)$.

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 $P(Z > 2)$.

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 $P(Z > z) = 0.10$ ($z \approx 1.28$), then $X = 500 + (1.28)(100) = 628$

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

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