Q: A hospital wants to set a cutoff for the lowest 3% of healthy human body temperatures, which are normally distributed with mean $$98.2^\circF $$and standard deviation $$0.73^\circF. $$What is the cutoff temperature?

$$x = 98.2 - 1.88 \times 0.73 = 96.83^\circF$

Q: If $$X \sim N(40, 6^2$)$$, what is the 95th percentile of $$X$$?

$$x = 40 + 6 \times 1.645 = 49.87$$

Q: A student scored in the 95th percentile on a standardized test. If the test scores are normally distributed with mean $$\mu = 100$$ and $$\sigma = 15$$, what is the student's score?

$$100 + 1.645 \times 15 = 124.675$$

Question 1

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

Accepted Answer

Its mean is 0 and its standard deviation is 1.

Question 2

Suppose the heights of adult women in a country are normally distributed with mean $$\mu = 64.5$$ inches and standard deviation $$\sigma = 2.5$$ inches. According to the empirical rule, approximately what percentage of women are between 62 inches and 67 inches tall?

Accepted Answer

68%

Question 3

Given a normal random variable $$X \sim N(100, 15^2$)$$, what is the $$z-$$score for $$x = 120$$?

Accepted Answer

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

Question 4

If the area under the standard normal curve to the left of $$z = 1.33 is$$ $$0.9082$$, what is the probability that $$Z \u003e 1.33$$?

Accepted Answer

$$0.0918$$

Question 5

A sixth-grade class has heights normally distributed with $$\mu = 55$$ inches and $$\sigma = 2.5$$ inches. Billy is 50 inches tall. What is his $$z-$$score?

Accepted Answer

$$z = \dfrac{50 - 55}{2.5} = -2$$

Question 6

Which property is shared by all normal distributions?

Accepted Answer

The total area under the curve is equal to 1.

Question 7

For a standard normal distribution, what is the probability that $$Z$$ falls between $$-1$$ and $$1$$?

Accepted Answer

Approximately $$68\%$$

Question 8

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

Accepted Answer

$$x = 98.2 - 1.88 	imes 0.73 = 96.83^\circF$

Question 9

If $$X \sim N(40, 6^2$)$$, what is the 95th percentile of $$X$$?

Accepted Answer

$$x = 40 + 6 	imes 1.645 = 49.87$$

Question 10

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

Accepted Answer

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

Question 11

If the area under the standard normal curve to the left of $$z = -0.38 is$$ $$0.3520$$, what is the area to the right of $$z = -0.38$$?

Accepted Answer

$$0.6480$$

Question 12

A researcher wants to find the probability that a standard normal variable $$Z$$ falls between $$z_1 = -1.02$ and z_2$ = 2.94. If $$the area to the left of $$z_2 is$$ $$0.9984$$ and the area to the left of $$z_1 is$$ $$0.1539$$, what is the probability?

Accepted Answer

$$0.9984 - 0.1539 = 0.8445$$

Question 13

Which of the following is TRUE about the probability of observing a specific value for a continuous random variable?

Accepted Answer

The probability is always 0.

Question 14

If $$P(Z \leq 1.68) = 0.9535$$, what is $$P(Z \u003e 1.68)$$?

Accepted Answer

$$1 - 0.9535 = 0.0465$$

Question 15

A student scored in the 95th percentile on a standardized test. If the test scores are normally distributed with mean $$\mu = 100$$ and $$\sigma = 15$$, what is the student's score?

Accepted Answer

$$100 + 1.645 	imes 15 = 124.675$$

Question 16

Which of the following best describes the relationship between the mean, median, and mode in a normal distribution?

Accepted Answer

Mean = median = mode

Question 17

If $$X \sim N(10, 5^2$)$$, what is the probability that $$X is $$between 12 and 15?

Accepted Answer

$$P(0.4 \leq Z \leq 1.0) = P(Z \leq 1.0) - P(Z \leq 0.4) = 0.8413 - 0.6554 = 0.1859$$

Question 18

To find the $$z-$$score corresponding to the 95th percentile, what should you do if the value 0.95 is not in the body of the $$z-$$table?

Accepted Answer

Take the average of the $$z-$$scores corresponding to the two closest probabilities.

Study Notes: The Normal Probability Distribution and Standardization

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