Q: A company wants to guarantee their tires for a mileage such that only 2% of tires will wear out before the guaranteed mileage. If tread life is normally distributed with mean 32,000 miles and standard deviation 2,500 miles, what mileage should they guarantee?

$$32,000 - (2.05 \times 2,500) = 26,875$$ miles

Q: A person’s body temperature is at the 25th percentile of a normal distribution with mean 98.2°F and standard deviation 0.7°F. What is their approximate temperature?

$$98.2 + (-0.674 \times 0.7) \approx 97.7$$°F

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 distribution 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 condition for using the Normal model to describe a data set?

Accepted Answer

The data must be skewed to the right

Question 4

If a person’s body temperature is 100.0°F, and the mean is 98.2°F with a standard deviation of 0.7°F, what is the z-score for this temperature?

Accepted Answer

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

Question 5

What does a z-score of 0 represent in a standardized data set?

Accepted Answer

The value is equal to the mean

Question 6

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

Accepted Answer

About 96%

Question 7

Which of the following best describes the effect of converting all values in a data set to z-scores?

Accepted Answer

The mean becomes 0 and the standard deviation becomes 1

Question 8

A company wants to guarantee their tires for a mileage such that only 2% of tires will wear out before the guaranteed mileage. If tread life is normally distributed with mean 32,000 miles and standard deviation 2,500 miles, what mileage should they guarantee?

Accepted Answer

$$32,000 - (2.05 	imes 2,500) = 26,875$$ miles

Question 9

Which of the following statements about the 68-95-99.7 Rule is correct?

Accepted Answer

About 95% of data in a normal distribution falls within 2 standard deviations of the mean

Question 10

A histogram of CEO compensation is strongly skewed to the right. Is it appropriate to use the Normal model to describe this data?

Accepted Answer

No, because the data is not unimodal and symmetric

Question 11

If the p-value from a Shapiro-Wilk Goodness-of-Fit test is 0.2878, what is the correct conclusion at the 0.05 significance level?

Accepted Answer

Fail to reject the null hypothesis; the data is nearly normal

Question 12

Which graphical tool is most useful for checking if a data set is approximately normal?

Accepted Answer

Normal probability plot (Q-Q plot)

Question 13

A person’s body temperature is at the 25th percentile of a normal distribution with mean 98.2°F and standard deviation 0.7°F. What is their approximate temperature?

Accepted Answer

$$98.2 + (-0.674 	imes 0.7) \approx 97.7$$°F

Question 14

If a value has a z-score of 3.5 in a normal distribution, how would you describe this value?

Accepted Answer

It is an extreme outlier

Question 15

Which of the following is true about percentiles in a normal distribution?

Accepted Answer

The 50th percentile is the median

Question 16

If you convert a data set from Fahrenheit to Celsius, what happens to the z-scores?

Accepted Answer

The z-scores remain unchanged

Question 17

A normal probability plot of a data set is very straight. What does this indicate?

Accepted Answer

The data is approximately normal

Question 18

If a data set is standardized, what is the mean and standard deviation of the resulting z-scores?

Accepted Answer

Mean = 0, Standard deviation = 1

Question 19

A value is 1.5 standard deviations above the mean in a normal distribution. Approximately what percentile does this correspond to?

Accepted Answer

About the 93rd percentile

Chapter 5: The Standard Deviation as a Ruler and the Normal Model – Study Notes

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