Q: Suppose the temperature of a solution is normally distributed with $$\mu = 25^\circC $$and $$\sigma = 2^\circC. $$What temperature corresponds to the 97.72th percentile? (Use $$Z = 2.0$$ for 97.72th percentile)

$$25 + 2 \times 2 = 29^\circC$$

Q: A company wants to know the value of $$X$$ such that only 5% of items are thicker than $$X. If $$thickness is normally distributed with $$\mu = 2.0 mm $$and $$\sigma = 0.1 mm$$, and $$P(Z \u003e 1.645) = 0.05$$, what is $$X$$?

$$X = 2.0 + 1.645 \times 0.1 = 2.1645 mm$$

Question 1

Which of the following is NOT a characteristic of the normal distribution?

Accepted Answer

The mean, median, and mode are all equal.

Question 2

A continuous variable is best described as:

Accepted Answer

A variable that can assume any value on a continuum, limited only by measurement precision.

Question 3

Given a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the Z-score for $$X = 140$$?

Accepted Answer

$$Z = \frac{140 - 100}{20} = 2$$

Question 4

Which of the following statements about the standardized normal distribution (Z distribution) is TRUE?

Accepted Answer

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

Question 5

Suppose the time to download a file is normally distributed with mean $$\mu = 18.0$$ seconds and standard deviation $$\sigma = 5.0$$ seconds. What is the probability that a randomly selected download takes less than 18.6 seconds? (Use the Z-table: $$P(Z \u003c 0.12) = 0.5478)$$

Accepted Answer

0.5478

Question 6

For a normal distribution, what is the total area under the curve?

Accepted Answer

1.0

Question 7

If $$P(Z  0.12)$$?

Accepted Answer

0.4522

Question 8

Suppose $$X is $$normally distributed with $$\mu = 18.0$$ and $$\sigma = 5.0. $$What is $$P(18 \u003c X \u003c 18.6)$$? (Given $$P(Z \u003c 0.12) = 0.5478$$ and $$P(Z \u003c 0) = 0.5000)$$

Accepted Answer

0.0478

Question 9

Which of the following best describes the process of standardizing a normal random variable $$X$$?

Accepted Answer

Subtract the mean from $$X$$ and divide by the standard deviation: $$Z = \frac{X - \mu}{\sigma}$$

Question 10

A company finds that the thickness of a manufactured item is normally distributed with mean $$\mu = 2.0 mm $$and standard deviation $$\sigma = 0.1 mm. $$What is the probability that a randomly selected item has a thickness greater than 2.15 mm? (Use $$P(Z \u003c 1.5) = 0.9332)$$

Accepted Answer

0.0668

Question 11

If the mean, median, and mode of a distribution are all equal, and the distribution is symmetric and bell-shaped, which type of distribution is this?

Accepted Answer

Normal distribution

Question 12

Suppose the temperature of a solution is normally distributed with $$\mu = 25^\circC $$and $$\sigma = 2^\circC. $$What temperature corresponds to the 97.72th percentile? (Use $$Z = 2.0$$ for 97.72th percentile)

Accepted Answer

$$25 + 2 	imes 2 = 29^\circC$$

Question 13

A Z-score of -1.5 indicates that the value is:

Accepted Answer

1.5 standard deviations below the mean

Question 14

If $$P(Z \u003c a) = 0.2000$$, what is the value of $$a$$? (Use the Z-table: $$P(Z \u003c -0.84) = 0.2000)$$

Accepted Answer

$$a = -0.84$$

Question 15

A hospital records the time to complete a certain medical test as normally distributed with mean $$\mu = 30$$ minutes and standard deviation $$\sigma = 4$$ minutes. What is the probability that a randomly selected test takes between 30 and 34 minutes? (Use $$P(Z \u003c 1) = 0.8413$$ and $$P(Z \u003c 0) = 0.5000)$$

Accepted Answer

0.3413

Question 16

Which of the following is a real-world example of a continuous variable?

Accepted Answer

Temperature of a chemical solution

Question 17

If the area to the left of $$Z = 0.5 is 0.6915$$, what is the probability that $$Z is $$greater than 0.5?

Accepted Answer

0.3085

Question 18

A company wants to know the value of $$X$$ such that only 5% of items are thicker than $$X. If $$thickness is normally distributed with $$\mu = 2.0 mm $$and $$\sigma = 0.1 mm$$, and $$P(Z \u003e 1.645) = 0.05$$, what is $$X$$?

Accepted Answer

$$X = 2.0 + 1.645 	imes 0.1 = 2.1645 mm$$

Question 19

Which of the following best explains why the probability of any single, exact value in a continuous distribution is zero?

Accepted Answer

Because the area under the curve at a single point is zero.

The Normal Distribution and Other Continuous Distributions

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