Q: The heights of adult women are normally distributed with a mean of 160 cm and a standard deviation of 7 cm. What is the height $$x$$ such that 5% of women are shorter than $$x$$?

$$x = 160 + (-1.64) \times 7 = 148.52 cm$$

Q: If $$X is a $$normal random variable with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what value of $$X$$ corresponds to the 95th percentile?

$$X = 50 + 1.64 \times 10 = 66.4$$

Q: A researcher wants to find the value $$x$$ such that only 2.5% of values in a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$ are less than $$x. $$What is $$x$$?

$$x = 100 + (-1.96) \times 15 = 70.6$$

Question 1

A bus arrives at a stop with arrival times normally distributed with a mean of 12 minutes and a standard deviation of 3 minutes. What is the probability that the bus arrives in less than 10 minutes?

Accepted Answer

$$P(X \u003c 10) = P\left(Z \u003c \frac{10-12}{3}\right) = P(Z \u003c -0.67) \approx 0.2514$$

Question 2

The commute times for a group of people are normally distributed with a mean of 20 minutes and a standard deviation of 5 minutes. What is the probability that a randomly selected person commutes for less than 15 minutes?

Accepted Answer

$$P(X \u003c 15) = P\left(Z \u003c \frac{15-20}{5}\right) = P(Z \u003c -1) \approx 0.1587$$

Question 3

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

Accepted Answer

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

Question 4

The heights of adult women are normally distributed with a mean of 160 cm and a standard deviation of 7 cm. What is the height $$x$$ such that 5% of women are shorter than $$x$$?

Accepted Answer

$$x = 160 + (-1.64) 	imes 7 = 148.52 cm$$

Question 5

A company finds that the time to complete a task is normally distributed with a mean of 30 minutes and a standard deviation of 4 minutes. What is the probability that a randomly selected task takes between 28 and 36 minutes?

Accepted Answer

$$P(28 \u003c X \u003c 36) = P\left(\frac{28-30}{4} \u003c Z \u003c \frac{36-30}{4}\right) = P(-0.5 \u003c Z \u003c 1.5) = 0.7745 - 0.3085 = 0.4660$$

Question 6

If $$X is a $$normal random variable with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what value of $$X$$ corresponds to the 95th percentile?

Accepted Answer

$$X = 50 + 1.64 	imes 10 = 66.4$$

Question 7

A student’s test scores are normally distributed with a mean of 75 and a standard deviation of 8. What is the $$z-$$score for a test score of 90?

Accepted Answer

$$z = \frac{90 - 75}{8} = 1.875$$

Question 8

A hospital records the recovery times for a certain procedure as normally distributed with a mean of 6 days and a standard deviation of 1.5 days. What is the probability that a patient recovers in more than 8 days?

Accepted Answer

$$P(X \u003e 8) = P\left(Z \u003e \frac{8-6}{1.5}\right) = P(Z \u003e 1.33) = 1 - 0.9082 = 0.0918$$

Question 9

Which of the following statements about $$z-$$scores is TRUE?

Accepted Answer

A $$z-$$score indicates how many standard deviations a data point is from the mean.

Question 10

A researcher wants to find the value $$x$$ such that only 2.5% of values in a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$ are less than $$x. $$What is $$x$$?

Accepted Answer

$$x = 100 + (-1.96) 	imes 15 = 70.6$$

Question 11

The arrival times of a bus are normally distributed with mean $$\mu = 12$$ min and standard deviation $$\sigma = 3$$ min. What is the probability that the bus arrives between 11 and 20 minutes late?

Accepted Answer

$$P(11 \u003c X \u003c 20) = P\left(\frac{11-12}{3} \u003c Z \u003c \frac{20-12}{3}\right) = P(-0.33 \u003c Z \u003c 2.67) = 0.9962 - 0.3707 = 0.6255$$

Question 12

A random variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the $$z-$$score for $$X = 40$$?

Accepted Answer

$$z = \frac{40 - 50}{10} = -1$$

Question 13

A population’s cholesterol levels are normally distributed with mean $$\mu = 200 mg/dL $$and standard deviation $$\sigma = 25 mg/dL. $$What cholesterol level corresponds to the top 1% of the population?

Accepted Answer

$$x = 200 + 2.33 	imes 25 = 258.25 mg/dL$$

Question 14

A manufacturer finds that the weights of its products are normally distributed with mean $$\mu = 500 g $$and standard deviation $$\sigma = 20 g. $$What is the probability that a product weighs more than 540 g?

Accepted Answer

$$P(X \u003e 540) = P\left(Z \u003e \frac{540-500}{20}\right) = P(Z \u003e 2) = 1 - 0.9772 = 0.0228$$

Question 15

Which of the following best describes the process of finding a value $$x$$ from a given probability in a non-standard normal distribution?

Accepted Answer

Find the corresponding $$z-$$score from the standard normal table, then use $$x = \mu + z \sigma.$$

Question 16

A city’s daily rainfall amounts are normally distributed with mean $$\mu = 5 mm $$and standard deviation $$\sigma = 2 mm. $$What is the probability that a randomly selected day has rainfall between 3 mm and 7 mm?

Accepted Answer

$$P(3 \u003c X \u003c 7) = P\left(\frac{3-5}{2} \u003c Z \u003c \frac{7-5}{2}\right) = P(-1 \u003c Z \u003c 1) = 0.8413 - 0.1587 = 0.6826$$

Question 17

A student’s exam scores are normally distributed with mean $$\mu = 70$$ and standard deviation $$\sigma = 10. $$What score corresponds to the lowest 10% of the distribution?

Accepted Answer

$$x = 70 + (-1.28) 	imes 10 = 57.2$$

Non-Standard Normal Distributions: Z-Scores and Probability Applications

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