Q: A tire company wants to guarantee their tires for only the top 2% of tread life. If tread life is normally distributed with mean $$32,000$$ miles and standard deviation $$2,500$$ miles, what is the minimum mileage they should guarantee?

$$32,000 + 2.05 \times 2,500 = 37,125$$ miles

Question 1

Suppose the heights of adult men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. What is the z-score for a man who is 75 inches tall?

Accepted Answer

$$z = \frac{75 - 69}{3} = 2$$

Question 2

A tire company claims the tread life of its snow tires follows a normal model with mean $$\mu = 32,000$$ miles and standard deviation $$\sigma = 2,500$$ miles. What percent of tires are expected to last between 29,500 and 34,500 miles?

Accepted Answer

About 68%, according to the 68-95-99.7 Rule

Question 3

Which of the following is NOT a property of z-scores after standardizing a data set?

Accepted Answer

The shape of the distribution always becomes perfectly normal

Question 4

A person has a body temperature of 100.0°F. If the mean body temperature is 98.2°F with a standard deviation of 0.7°F, what is this person's z-score?

Accepted Answer

$$z = \frac{100.0 - 98.2}{0.7} \approx 2.57$$

Question 5

Which of the following best describes the 68-95-99.7 Rule for a normal distribution?

Accepted Answer

About 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean

Question 6

If a distribution is strongly skewed to the right, which of the following is the best course of action before applying a normal model?

Accepted Answer

Do not use the normal model, as the Nearly Normal Condition is not met

Question 7

A student scores 680 on the SAT, which is normally distributed with mean 500 and standard deviation 100. What percent of students scored below this student?

Accepted Answer

About 96%

Question 8

Which of the following statements about percentiles is TRUE?

Accepted Answer

The 25th percentile is the value below which 25% of the data falls

Question 9

A tire company wants to guarantee their tires for only the top 2% of tread life. If tread life is normally distributed with mean $$32,000$$ miles and standard deviation $$2,500$$ miles, what is the minimum mileage they should guarantee?

Accepted Answer

$$32,000 + 2.05 	imes 2,500 = 37,125$$ miles

Question 10

Which of the following is the correct formula for calculating a z-score?

Accepted Answer

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

Question 11

A normal probability plot is used to:

Accepted Answer

Assess whether a data set is approximately normal

Question 12

If the p-value from a Goodness of Fit test for normality is 0.03, what is the correct conclusion at the 0.05 significance level?

Accepted Answer

Reject the null hypothesis; the data is not nearly normal

Question 13

Which of the following best describes what happens to the mean and standard deviation when a data set is standardized to z-scores?

Accepted Answer

The mean becomes 0 and the standard deviation becomes 1

Question 14

A histogram of CEO compensation is strongly skewed right. What does this suggest about using the normal model for this data?

Accepted Answer

The normal model is not appropriate because the data is not unimodal and symmetric

Question 15

If a value has a z-score of 3.2 in a normal distribution, how would you describe this value?

Accepted Answer

It is an unusual value and likely an outlier

Question 16

Which of the following is a correct interpretation of the 95th percentile in a normal distribution?

Accepted Answer

95% of the data falls below this value

Question 17

If the mean daily high temperature in Fahrenheit is 77.2°F with a standard deviation of 5.83°F, what is the z-score for a day with a high of 87°F?

Accepted Answer

$$z = \frac{87 - 77.2258}{5.8293} \approx 1.68$$

Question 18

Which of the following is the best definition of a percentile?

Accepted Answer

A value below which a given percentage of observations fall

The Standard Deviation as a Ruler and the Normal Model

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