Q: A company wants to design a doorway so that 95% of adults can walk through without bending. If adult heights are normally distributed with $$\mu = 66.2 in. $$and $$\sigma = 3.8 in$$., what doorway height should they choose? (Use $$z = 1.645$$ for the 95th percentile.)

$$x = 66.2 + 1.645 \times 3.8 = 72.45 in.$$

Q: A company wants to design a cockpit to accommodate the shortest 95% of women. If women's heights are normally distributed with $$\mu = 63.7 in. $$and $$\sigma = 2.9 in$$., what is the maximum allowable height for the cockpit? (Use $$z = 1.645.)$$

$$x = 63.7 + 1.645 \times 2.9 = 68.47 in.$$

Q: A normal distribution has $$\mu = 100$$ and $$\sigma = 15. $$What value of $$x$$ corresponds to the 97.5th percentile? (Use $$z = 1.96.)$$

$$x = 100 + 1.96 \times 15 = 129.4$$

Question 1

Which formula is used to convert a value $$x$$ from a nonstandard normal distribution with mean $$\mu$$ and standard deviation $$\sigma to a $$standard normal $$z-$$score?

Accepted Answer

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

Question 2

Suppose the heights of adult men are normally distributed with a mean of 68.6 inches and a standard deviation of 2.8 inches. What is the $$z-$$score for a man who is 72 inches tall?

Accepted Answer

$$z = \frac{72 - 68.6}{2.8} = 1.21$$

Question 3

If the area to the left of $$z = 1.21 in $$the standard normal distribution is 0.8869, what is the probability that a randomly selected man is taller than 72 inches (given $$\mu = 68.6$$, $$\sigma = 2.8$$)?

Accepted Answer

$$1 - 0.8869 = 0.1131$$

Question 4

What are the mean and standard deviation of a distribution after all values are converted to $$z-$$scores using $$z = \frac{x - \mu}{\sigma}$$?

Accepted Answer

Mean = 0, Standard deviation = 1

Question 5

A company wants to design a doorway so that 95% of adults can walk through without bending. If adult heights are normally distributed with $$\mu = 66.2 in. $$and $$\sigma = 3.8 in$$., what doorway height should they choose? (Use $$z = 1.645$$ for the 95th percentile.)

Accepted Answer

$$x = 66.2 + 1.645 	imes 3.8 = 72.45 in.$$

Question 6

What is the main difference between a standard normal distribution and a nonstandard normal distribution?

Accepted Answer

The mean and standard deviation

Question 7

If the area to the left of a value $$x in a $$normal distribution is 0.9500, what is the corresponding $$z-$$score?

Accepted Answer

$$z = 1.645$$

Question 8

A hospital wants to set a minimum birth weight requirement so that only the lightest 10% of newborns are flagged for special care. If birth weights are normally distributed with $$\mu = 3152.0 g $$and $$\sigma = 693.4 g$$, what $$z-$$score should be used to find this cutoff?

Accepted Answer

$$z = -1.28$$

Question 9

Which of the following steps is NOT part of the procedure for finding the value $$x$$ from a known area (probability) under a normal curve?

Accepted Answer

Multiply the area by the standard deviation

Question 10

A company wants to design a cockpit to accommodate the shortest 95% of women. If women's heights are normally distributed with $$\mu = 63.7 in. $$and $$\sigma = 2.9 in$$., what is the maximum allowable height for the cockpit? (Use $$z = 1.645.)$$

Accepted Answer

$$x = 63.7 + 1.645 	imes 2.9 = 68.47 in.$$

Question 11

If a $$z-$$score is negative, where is the corresponding $$x$$ value located on the normal distribution?

Accepted Answer

To the left of the mean

Question 12

Which of the following statements about areas under the normal curve is TRUE?

Accepted Answer

Areas (probabilities) are always between 0 and 1

Question 13

A value $$x$$ separates the top 10% from the rest in a normal distribution. On which side of the mean is $$x$$ located?

Accepted Answer

Right side

Question 14

If you know the $$z-$$score and want to find the corresponding $$x$$ value in a normal distribution, which formula should you use?

Accepted Answer

$$x = \mu + z \cdot \sigma$$

Question 15

A normal distribution has $$\mu = 100$$ and $$\sigma = 15. $$What value of $$x$$ corresponds to the 97.5th percentile? (Use $$z = 1.96.)$$

Accepted Answer

$$x = 100 + 1.96 	imes 15 = 129.4$$

Study Guide: Real Applications of Normal Probability Distributions

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