Q: A manufacturer wants to design a doorway that will accommodate at least 95% of adult heights, which are normally distributed with a mean of 66.2 inches and a standard deviation of 3.8 inches. What is the minimum height the door should be, using the z-score that corresponds to the 95th percentile?

$$x = 66.2 + (1.645 \times 3.8) = 72.45$$ inches

Question 1

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 2

A manufacturer wants to design a doorway that will accommodate at least 95% of adult heights, which are normally distributed with a mean of 66.2 inches and a standard deviation of 3.8 inches. What is the minimum height the door should be, using the z-score that corresponds to the 95th percentile?

Accepted Answer

$$x = 66.2 + (1.645 	imes 3.8) = 72.45$$ inches

Question 3

If the area to the left of a certain z-score is 0.9500, what is the value of the z-score?

Accepted Answer

$$z = 1.645$$

Question 4

Which of the following statements about z-scores and areas under the normal curve is TRUE?

Accepted Answer

Areas (probabilities) are always between 0 and 1, and z-scores can be positive or negative.

Question 5

A company wants to know what proportion of men are taller than a showerhead set at 72 inches. Heights are normally distributed with mean 68.6 inches and standard deviation 2.8 inches. What is the correct procedure to find this proportion?

Accepted Answer

Convert 72 inches to a z-score, find the cumulative area to the left using the z-table, then subtract this area from 1.

Question 6

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

Accepted Answer

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

Question 7

If you are given a probability (area) and need to find the corresponding $$x$$ value in a normal distribution, which steps should you follow?

Accepted Answer

Find the z-score corresponding to the area, then use $$x = \mu + z\sigma to $$solve for $$x.$$

Question 8

A value separates the top 10% from the rest in a normal distribution. Where is this value located on the normal curve?

Accepted Answer

On the right side of the graph.

Question 9

If a z-score is negative, what does this indicate about the corresponding $$x$$ value in a normal distribution?

Accepted Answer

The $$x$$ value is less than the mean.

Question 10

A hospital finds that the weights of newborns are normally distributed with a mean of 7.5 lbs and a standard deviation of 1.2 lbs. What is the probability that a randomly selected newborn weighs less than 6 lbs? (Use the z-score formula and the z-table area for your answer.)

Accepted Answer

First, $$z = \frac{6 - 7.5}{1.2} = -1.25. $$Then, use the z-table to find the area to the left of $$z = -1.25.$$

Question 11

Which of the following is a correct interpretation of a cumulative area from the left in a standard normal distribution?

Accepted Answer

It represents the probability that a randomly selected value is less than or equal to a given $$z$$ score.

Question 12

A value $$x in a $$normal distribution corresponds to a cumulative area of 0.10 from the left. What can you say about the z-score for $$x$$?

Accepted Answer

The z-score is negative.

Question 13

Why is it important to sketch a normal curve and shade the relevant region when solving normal distribution problems?

Accepted Answer

It helps visualize and correctly identify the area or probability being sought.

Question 14

If the area to the left of $$x is 0.80 in a $$normal distribution, what is the percentile rank of $$x$$?

Accepted Answer

80th percentile

Question 15

A value $$x$$ separates the bottom 5% of a normal distribution. What is the approximate z-score for $$x$$?

Accepted Answer

$$z \approx -1.645$$

Question 16

Which of the following is NOT a correct property of areas under the normal curve?

Accepted Answer

Areas can be negative if the z-score is negative.

Question 17

A student scores 85 on a test where the mean is 80 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 80}{5} = 1$$

Question 18

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

Accepted Answer

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

Question 19

A city wants to set a minimum height requirement for a ride so that only the tallest 10% of children are excluded. If children's heights are normally distributed, what should the city do to determine the cutoff height?

Accepted Answer

Find the z-score corresponding to the 90th percentile, then use $$x = \mu + z\sigma to $$calculate the cutoff.

Normal Probability Distributions: Standardization, Area Calculation, and Real-World Applications

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