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 is the minimum doorway height required?

$$x = 66.2 + 1.645 \times 3.8 = 72.5$$\text{ in.}$

Q: A university wants to award scholarships to students scoring in the top 5% on a standardized test. If test scores are normally distributed with $$\mu = 500$$ and $$\sigma = 100$$, what is the minimum score required for a scholarship?

$$x = 500 + 1.645 \times 100 = 664.5 ($$since top 5% corresponds to $$z = 1.645)$$

Question 1

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

Accepted Answer

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

Question 2

If the area to the left of a certain $$z-$$score is $$0.8869$$, what is the probability that a randomly selected value from this normal distribution is greater than the corresponding $$x$$ value?

Accepted Answer

$$1 - 0.8869 = 0.1131$$

Question 3

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 is the minimum doorway height required?

Accepted Answer

$$x = 66.2 + 1.645 	imes 3.8 = 72.5$$	ext{ in.}$

Question 4

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

Accepted Answer

It has a mean of $$0$$ and a standard deviation of $$1.$$

Question 5

Given a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, which formula correctly converts an $$x$$ value to a $$z-$$score?

Accepted Answer

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

Question 6

If the area to the left of $$z is$$ $$0.9500$$, what is the corresponding $$z-$$score (rounded to three decimal places)?

Accepted Answer

$$z = 1.645$$

Question 7

A hospital wants to install showerheads so that only 10% of men are taller than the showerhead. If men's heights are normally distributed with $$\mu = 68.6 in. $$and $$\sigma = 2.8 in$$., what $$z-$$score should be used to find the required height?

Accepted Answer

$$z = 1.28 ($$since the area to the left is 0.90, so $$z = 1.28)$$

Question 8

Which of the following is NOT a correct step when finding the value $$x$$ from a known area under a normal curve?

Accepted Answer

Use the area directly as the $$z-$$score in the formula $$x = \mu + z \cdot \sigma.$$

Question 9

If a value $$x is $$located in the left half of a normal distribution, what can be said about its $$z-$$score?

Accepted Answer

The $$z-$$score is negative.

Question 10

A university wants to award scholarships to students scoring in the top 5% on a standardized test. If test scores are normally distributed with $$\mu = 500$$ and $$\sigma = 100$$, what is the minimum score required for a scholarship?

Accepted Answer

$$x = 500 + 1.645 	imes 100 = 664.5 ($$since top 5% corresponds to $$z = 1.645)$$

Question 11

Which of the following best describes the relationship between $$z-$$scores and areas under the normal curve?

Accepted Answer

$$z-$$scores are distances along the horizontal axis, while areas are regions under the curve.

Question 12

If the cumulative area to the left of $$z is$$ $$0.10$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = -1.28$$

Question 13

A researcher finds that the probability of a randomly selected value being less than $$x is$$ $$0.975. $$What is the corresponding $$z-$$score?

Accepted Answer

$$z = 1.96$$

Question 14

Which Excel function can be used to find the $$x$$ value corresponding to a given cumulative probability in a normal distribution?

Accepted Answer

NORM.INV

Question 15

If the total area under a normal curve is 1, what does an area of 0.1123 to the right of $$x = 72 in. ($$with $$\mu = 68.6$$, $$\sigma = 2.8) $$represent?

Accepted Answer

The proportion of men taller than 72 inches.

Question 16

Which of the following is a correct interpretation of a $$z-$$score of $$1.21$$?

Accepted Answer

The value is $$1.21$$ standard deviations above the mean.

6.2 Normal Probability Distributions: Real Applications and Methods

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