Q: For aircraft cabin design, what ceiling height will accommodate $$95\% of $$men if men’s heights are normally distributed with mean $$69.5$$ inches and standard deviation $$2.4$$ inches?

$$69.5 + 1.645 \times 2.4 = 73.45$$ inches

Question 1

A bone mineral density test result is measured as a $$z$$ score. If the population of $$z$$ scores is normally distributed with mean $$0$$ and standard deviation $$1$$, what is the probability that a randomly selected adult has a bone density test score less than $$1.27$$?

Accepted Answer

Approximately $$0.8980$$

Question 2

For the same bone density test, what is the probability that a randomly selected person has a result above $$-1.00$$?

Accepted Answer

Approximately $$0.8413$$

Question 3

A bone density reading between $$-2.50$$ and $$-1.00$$ indicates osteopenia. What is the probability that a randomly selected adult has osteopenia?

Accepted Answer

Approximately $$0.1066$$

Question 4

Using the standard normal distribution, what $$z$$ score corresponds to the $$95th $$percentile ($$P_{95}$$)?

Accepted Answer

$$z \approx 1.645$$

Question 5

What $$z$$ scores separate the bottom $$2.5\%$$ and the top $$2.5\% of $$the standard normal distribution?

Accepted Answer

$$z \approx -1.96$$ and $$z \approx 1.96$$

Question 6

Tall Clubs International requires women to be at least $$70$$ inches tall. If women’s heights are normally distributed with mean $$63.8$$ inches and standard deviation $$2.6$$ inches, what percentage of women meet this requirement?

Accepted Answer

Approximately $$2.28\%$$

Question 7

For aircraft cabin design, what ceiling height will accommodate $$95\% of $$men if men’s heights are normally distributed with mean $$69.5$$ inches and standard deviation $$2.4$$ inches?

Accepted Answer

$$69.5 + 1.645 	imes 2.4 = 73.45$$ inches

Question 8

If a bone density test score is at the $$5th $$percentile, what is the corresponding $$z$$ score?

Accepted Answer

$$z \approx -1.645$$

Question 9

A woman’s height is $$70$$ inches. What is her $$z$$ score if the population mean is $$63.8$$ inches and the standard deviation is $$2.6$$ inches?

Accepted Answer

$$z = \frac{70 - 63.8}{2.6} \approx 2.38$$

Question 10

If the probability that a randomly selected adult has a bone density score less than $$z is$$ $$0.5$$, what is the value of $$z$$?

Accepted Answer

$$z = 0$$

Question 11

Which of the following best describes the area under the normal curve for $$z \u003c -1.00$$?

Accepted Answer

Approximately $$0.1587$$

Question 12

If a bone density test score is at the $$97.5th $$percentile, what is the corresponding $$z$$ score?

Accepted Answer

$$z \approx 1.96$$

Question 13

A bone density test result is considered 'normal' if the $$z$$ score is above $$-1.00. $$What percentage of adults have 'normal' bone density according to the standard normal distribution?

Accepted Answer

Approximately $$84.13\%$$

Question 14

If the area under the standard normal curve between $$z = -2.50$$ and $$z = -1.00 is$$ $$0.1066$$, what does this represent in the context of bone density tests?

Accepted Answer

The probability that a randomly selected adult has osteopenia

Question 15

Which formula is used to convert a raw score $$X to a$$ $$z$$ score in a normal distribution?

Accepted Answer

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

Question 16

If the mean height of men is $$69.5$$ inches and the standard deviation is $$2.4$$ inches, what is the $$z$$ score for a man who is $$74.3$$ inches tall?

Accepted Answer

$$z = \frac{74.3 - 69.5}{2.4} \approx 2.00$$

Question 17

A bone density score that separates the bottom $$95\%$$ from the top $$5\% is $$called the:

Accepted Answer

95th percentile

Question 18

If you want to find the probability that a value is between two $$z$$ scores in StatCrunch, which calculator option should you use?

Accepted Answer

Between

Question 19

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

Accepted Answer

The mean is $$0$$ and the standard deviation is $$1$$

Question 20

If the area under the standard normal curve to the left of $$z = 1.27 is$$ $$0.8980$$, what is the area to the right of $$z = 1.27$$?

Accepted Answer

$$1 - 0.8980 = 0.1020$$

Step-by-Step Guidance: Normal Distributions & Probabilities in Statistics

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