Q: If the mean height of female college students is 64.4 inches with a standard deviation of 2.4 inches, what height marks the 99th percentile?

$$x = 64.4 + 2.33 \times 2.4 = 69.99$$ inches

Question 1

Which of the following statements best describes what the area under the curve of a normal distribution represents?

Accepted Answer

Relative frequencies, percentages expressed as decimals, or probabilities.

Question 2

Suppose the heights of female college students are normally distributed with mean $$\mu = 64.4$$ inches and standard deviation $$\sigma = 2.4$$ inches. What is the z-score for a student who is 68 inches tall?

Accepted Answer

$$z = \frac{68 - 64.4}{2.4} = 1.5$$

Question 3

If you want to find the percentage of students shorter than 68 inches, which of the following steps is correct?

Accepted Answer

Find the z-score for 68 inches, then look up the area to the left of this z-score in the standard normal table.

Question 4

Given $$z = -0.58$$ for a height of 63 inches, what is the probability that a randomly selected student is taller than 63 inches?

Accepted Answer

$$1 - 0.2810 = 0.7190$$

Question 5

What is the formula to convert a z-score back to an x-value (destandardize) in a normal distribution?

Accepted Answer

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

Question 6

If the area to the left of a z-score is 0.94, what percentage of the distribution is to the right of this z-score?

Accepted Answer

6%

Question 7

A researcher wants to find the height that separates the tallest 6% of female college students from the rest. Which calculator function and parameters should they use?

Accepted Answer

Invnorm$$(0.94, 64.4, 2.4)$$

Question 8

Why are normal distributions so common in nature and social phenomena?

Accepted Answer

Because many random factors added together, regardless of their individual distributions, tend to form a normal distribution.

Question 9

If you are given a probability and want to find the corresponding x-value in a normal distribution, what is the correct sequence of steps?

Accepted Answer

Find the z-score from the table, then destandardize using $$x = \mu + z \sigma$$

Question 10

Which calculator function would you use to find the percentage of students with heights between 63 and 68 inches, given $$\mu = 64.4$$ and $$\sigma = 2.4$$?

Accepted Answer

Normalcdf$$(63, 68, 64.4, 2.4)$$

Question 11

A histogram of sample data appears bell-shaped. What additional step might a researcher take to justify using the normal distribution for analysis?

Accepted Answer

Perform a normal probability plot or other statistical test for normality.

Question 12

If the mean height of female college students is 64.4 inches with a standard deviation of 2.4 inches, what height marks the 99th percentile?

Accepted Answer

$$x = 64.4 + 2.33 	imes 2.4 = 69.99$$ inches

Question 13

Which of the following best describes the process of 'standardizing' a value in a normal distribution?

Accepted Answer

Subtract the mean and divide by the standard deviation: $$z = \frac{x - \mu}{\sigma}$$

Question 14

If you want to find the area between two x-values in a normal distribution using a calculator, which function should you use?

Accepted Answer

Normalcdf$$(a, b, \mu, \sigma)$$

Question 15

A student claims that the area under the standard normal curve to the left of $$z = 0 is 0.5. Is $$this correct?

Accepted Answer

Yes, because the standard normal distribution is symmetric about the mean.

The Normal Distribution: Concepts, Calculations, and Applications

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