Q: A standardized test has scores that are normally distributed with mean $$\mu = 500$$ and standard deviation $$\sigma = 100. $$What score corresponds to the 95th percentile?

$$x = 500 + 100 \times 1.645 = 664.5$$

Q: A hospital records body temperatures of patients, which are normally distributed with mean $$98.2^\circ F$$ and standard deviation $$0.73^\circ F. $$What is the cutoff temperature for the lowest 3% of healthy human body temperatures?

$$x = 98.2 + 0.73 \times (-1.88) = 96.83^\circ F$$

Q: A student scores in the 5th percentile on a test with mean $$\mu = 40$$ and standard deviation $$\sigma = 6. $$What is the student's score?

$$x = 40 + 6 \times (-1.645) = 30.13$$

Question 1

Suppose the heights of adult males 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

A standardized test has scores that are normally distributed with mean $$\mu = 500$$ and standard deviation $$\sigma = 100. $$What score corresponds to the 95th percentile?

Accepted Answer

$$x = 500 + 100 	imes 1.645 = 664.5$$

Question 3

If $$X \sim N(100, 16)$$, what is the probability that $$X is $$between 96 and 104?

Accepted Answer

$$P(96 \leq X \leq 104) = P(-1 \leq Z \leq 1) = P(Z \u003c 1) - P(Z \u003c -1) = 0.8413 - 0.1587 = 0.6826$$

Question 4

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

Accepted Answer

$$x = 98.2 + 0.73 	imes (-1.88) = 96.83^\circ F$$

Question 5

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

Accepted Answer

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

Question 6

Given $$X \sim N(10, 25)$$, what is $$P(X \leq 12)$$?

Accepted Answer

$$P(X \leq 12) = P(Z \leq 0.4) = 0.6554$$

Question 7

If $$Z is a $$standard normal random variable, what is $$P(-1.35 \u003c Z \u003c 2.01)$$?

Accepted Answer

$$P(-1.35 \u003c Z \u003c 2.01) = P(Z \u003c 2.01) - P(Z \u003c -1.35) = 0.9778 - 0.0885 = 0.8893$$

Question 8

A student scores in the 5th percentile on a test with mean $$\mu = 40$$ and standard deviation $$\sigma = 6. $$What is the student's score?

Accepted Answer

$$x = 40 + 6 	imes (-1.645) = 30.13$$

Question 9

If $$X \sim N(10, 25)$$, what is $$P(12 \leq X \leq 15)$$?

Accepted Answer

$$P(12 \leq X \leq 15) = P(Z \leq 1) - P(Z \leq 0.4) = 0.8413 - 0.6554 = 0.1859$$

Question 10

Which formula is used to convert a normal random variable $$X to a $$standard normal variable $$Z$$?

Accepted Answer

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

Question 11

If $$P(Z \u003c z_1$) = 0.9495$$ and $$P(Z \u003c z_2$) = 0.9505$$, what is the $$z-$$score for the 95th percentile?

Accepted Answer

$$z_0$$ = \frac{1.64 + 1.65}{2} = 1.645$$

Question 12

A researcher wants to find the value $$x$$ such that $$P(X \geq x) = 0.7995$$ for $$X \sim N(40, 36). $$What is $$x$$?

Accepted Answer

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

Question 13

Which of the following best describes the use of the standard normal table (Table V)?

Accepted Answer

It provides the area under the standard normal curve to the left of a given $$z-$$score.

Question 14

If $$Z is a $$standard normal random variable, what is $$P(Z \u003e 1.68)$$ using the complement rule?

Accepted Answer

$$P(Z \u003e 1.68) = 1 - P(Z \u003c 1.68) = 1 - 0.9535 = 0.0465$$

Question 15

A company wants to award bonuses to employees who score in the top 10% on a performance test. If the test scores are normally distributed with mean $$\mu = 80$$ and standard deviation $$\sigma = 10$$, what is the minimum score required for a bonus?

Accepted Answer

$$x = 80 + 10 	imes 1.28 = 92.8$$

Question 16

Which of the following is the correct formula for finding the area between two $$z-$$scores $$z_1$$ and $$z_2$$?

Accepted Answer

$$P(z_1$ \u003c Z \u003c z_2$) = P(Z \u003c z_2$) - P(Z \u003c z_1$)$$

Question 17

If $$X \sim N(10, 25)$$, what is $$P(|X| \leq 15)$$?

Accepted Answer

Question 18

When using the standard normal table, how should you find the $$z-$$score for a percentile that is not exactly in the table but is between two closest values?

Accepted Answer

Take the average of the corresponding $$z-$$scores for the two closest percentiles.

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

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