Q: A medical test result is normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value corresponds to the 90th percentile?

$$x = \mu + z_{0.90} \sigma = 100 + 1.28 \times 15 = 119.2$$

Q: A standardized test score is normally distributed with mean 500 and standard deviation 100. What is the cutoff score for the top 2.5% of test takers?

$$x = 500 + 100 \times 1.96 = 696$$

Q: Suppose $$X \sim N(60, 25)$$ and you want to find the value $$x$$ such that $$P(X \u003c x) = 0.975. $$What is $$x$$?

$$x = 60 + 5 \times 1.96 = 69.8$$

Q: A researcher wants to find the cutoff value for the lowest 5% of scores on a normally distributed exam with mean 80 and standard deviation 6. What is the cutoff score?

$$x = 80 + 6 \times (-1.645) = 70.13$$

Question 1

Suppose the heights of adult males in a city are normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the probability that a randomly selected male is taller than 73 inches?

Accepted Answer

$$P(X \u003e 73) = 1 - P(Z \u003c 1) = 1 - 0.8413 = 0.1587$$

Question 2

Given a standard normal random variable $$Z$$, what is the area under the curve between $$z = -1.25$$ and $$z = 0.75$$?

Accepted Answer

$$P(-1.25 \u003c Z \u003c 0.75) = P(Z \u003c 0.75) - P(Z \u003c -1.25) = 0.7734 - 0.1056 = 0.6678$$

Question 3

A medical test result is normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value corresponds to the 90th percentile?

Accepted Answer

$$x = \mu + z_{0.90} \sigma = 100 + 1.28 	imes 15 = 119.2$$

Question 4

If $$X \sim N(50, 16)$$, what is the probability that $$X is $$less than 54?

Accepted Answer

$$P(X \u003c 54) = P\left(Z \u003c \frac{54-50}{4}\right) = P(Z \u003c 1) = 0.8413$$

Question 5

A standardized test score is normally distributed with mean 500 and standard deviation 100. What is the cutoff score for the top 2.5% of test takers?

Accepted Answer

$$x = 500 + 100 	imes 1.96 = 696$$

Question 6

Which of the following statements about continuous random variables is TRUE?

Accepted Answer

For any continuous random variable $$X$$, $$P(a \u003c X \u003c b) = P(a \leq X \u003c b) = P(a \u003c X \leq b) = P(a \leq X \leq b)$$

Question 7

If $$Z is a $$standard normal random variable, what is $$P(Z \u003e 2.10)$$?

Accepted Answer

$$P(Z \u003e 2.10) = 1 - P(Z \u003c 2.10) = 1 - 0.9821 = 0.0179$$

Question 8

A machine produces bolts with lengths that are normally distributed with mean 5 cm and standard deviation 0.1 cm. What proportion of bolts are between 4.9 cm and 5.1 cm?

Accepted Answer

$$P(4.9 \u003c X \u003c 5.1) = P(-1 \u003c Z \u003c 1) = 0.8413 - 0.1587 = 0.6826$$

Question 9

Which of the following is the correct formula for converting a value $$x$$ from a normal distribution $$N(\mu, \$$sigma^2$$) to a $$standard normal $$z-$$score?

Accepted Answer

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

Question 10

Suppose $$X \sim N(60, 25)$$ and you want to find the value $$x$$ such that $$P(X \u003c x) = 0.975. $$What is $$x$$?

Accepted Answer

$$x = 60 + 5 	imes 1.96 = 69.8$$

Question 11

A researcher wants to find the cutoff value for the lowest 5% of scores on a normally distributed exam with mean 80 and standard deviation 6. What is the cutoff score?

Accepted Answer

$$x = 80 + 6 	imes (-1.645) = 70.13$$

Question 12

If $$Z is a $$standard normal variable, which of the following is the correct way to find $$P(-z_0$ \u003c Z \u003c z_0$)$$?

Accepted Answer

$$P(-z_0$ \u003c Z \u003c z_0$) = P(Z \u003c z_0$) - P(Z \u003c -z_0$)$$

Question 13

A hospital records body temperatures that are normally distributed with mean $$98.2^\circF $$and standard deviation $$0.73^\circF. $$What is the cutoff temperature for the lowest 3% of healthy human body temperatures?

Accepted Answer

$$x = 98.2 + 0.73 	imes (-1.88) = 96.8^\circF$$

Question 14

If $$P(Z \u003c z_1$) = 0.25$$ and $$P(Z \u003c z_2$) = 0.75$$, what is the probability that $$Z$$ falls between $$z_1$$ and $$z_2$$?

Accepted Answer

$$P(z_1$ \u003c Z \u003c z_2$) = 0.75 - 0.25 = 0.5$$

Question 15

Which of the following best describes how to use the standard normal table (Table V) to find the area to the right of a given $$z$$ value?

Accepted Answer

Find the area to the left of $$z in $$the table, then subtract this value from 1.

Question 16

A variable $$X is $$normally distributed with mean $$\mu = 40$$ and variance $$\$$sigma^2$$ = 36. $$What is the value $$x_0$$ such that $$P(X \geq x_0$) = 0.7995$$?

Accepted Answer

$$x_0 = 40 + 6$$ 	imes (-0.84) = 34.96$$

Question 17

If you are given a percentile $$p$$ and need to find the corresponding value $$x$$ for a normal random variable $$X \sim N(\mu, \$$sigma^2$$)$$, which formula should you use?

Accepted Answer

$$x = \mu + z_p \sigma$$

Lecture 22: The Normal Distribution – Areas, Probabilities, and Percentiles

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