Q: A normal distribution has a mean of 50 and a standard deviation of 5. What is the $$x-$$value that corresponds to a $$z-$$score of $$-1.2$$?

$$x = 50 + (-1.2) \times 5 = 44$$

Question 1

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

Accepted Answer

The curve is skewed to the right.

Question 2

A manufacturer finds that the lifetimes of its light bulbs are normally distributed with a mean of 1200 hours and a standard deviation of 100 hours. What is the z-score for a bulb that lasts 1350 hours?

Accepted Answer

$$z = \frac{1350 - 1200}{100} = 1.5$$

Question 3

The standard normal distribution is defined as a normal distribution with which of the following parameters?

Accepted Answer

Mean $$\mu = 0$$, standard deviation $$\sigma = 1$$

Question 4

If the area to the left of $$z = 1.15$$ under the standard normal curve is 0.8749, what is the area to the right of $$z = 1.15$$?

Accepted Answer

$$1 - 0.8749 = 0.1251$$

Question 5

A set of test scores is normally distributed with a mean of 600 and a standard deviation of 20. What is the probability that a randomly selected score is less than 640?

Accepted Answer

First, calculate $$z = \frac{640 - 600}{20} = 2. $$The area to the left of $$z = 2 is $$approximately 0.9772.

Question 6

Which of the following statements about inflection points on a normal curve is TRUE?

Accepted Answer

They occur at $$x = \mu \pm \sigma.$$

Question 7

Suppose the weights of newborns are normally distributed with a mean of 3300 grams and a standard deviation of 500 grams. What is the probability that a randomly selected newborn weighs more than 3800 grams? (Use the standard normal table: area to the left of $$z = 1 is 0.8413.)$$

Accepted Answer

First, $$z = \frac{3800 - 3300}{500} = 1. $$Area to the right is $$1 - 0.8413 = 0.1587.$$

Question 8

A normal distribution has a mean of 50 and a standard deviation of 5. What is the $$x-$$value that corresponds to a $$z-$$score of $$-1.2$$?

Accepted Answer

$$x = 50 + (-1.2) 	imes 5 = 44$$

Question 9

Which of the following best describes the effect of increasing the standard deviation $$\sigma of a $$normal distribution while keeping the mean $$\mu$$ constant?

Accepted Answer

The curve becomes wider and flatter.

Question 10

If a value has a $$z-$$score of 2.5 in a normal distribution, which of the following is TRUE?

Accepted Answer

It is considered unusual, as it is more than 2 standard deviations from the mean.

Question 11

A student scored 1.25 standard deviations above the mean on a standardized test. What percentage of students scored below this student? (Use the standard normal table: area to the left of $$z = 1.25 is 0.8944.)$$

Accepted Answer

Approximately 89.44%

Question 12

Which of the following is the correct formula to convert a raw score $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma to a$$ $$z-$$score?

Accepted Answer

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

Question 13

If the area to the left of $$z = -0.90 is 0.1841$$, what is the probability that $$z is $$greater than $$-0.90$$?

Accepted Answer

$$1 - 0.1841 = 0.8159$$

Question 14

A hospital records the birth weights of newborns, which are normally distributed with a mean of 3300 grams and a standard deviation of 500 grams. What is the probability that a randomly selected newborn weighs between 2800 grams and 3800 grams? (Use the standard normal table: area to the left of $$z = 1 is 0.8413$$, area to the left of $$z = -1 is 0.1587.)$$

Accepted Answer

First, $$z_1$$ = \frac{2800 - 3300}{500} = -1$$, z_2$ = \frac{3800 - 3300}{500} = 1. $$Area between is $$0.8413 - 0.1587 = 0.6826.$$

Question 15

Which of the following best describes the relationship between the area under the standard normal curve and probability?

Accepted Answer

The area under the curve to the left of a $$z-$$score gives the probability that $$z is $$less than that value.

Question 16

A normal distribution has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected value is less than 85? (Use the standard normal table: area to the left of $$z = -1 is 0.1587.)$$

Accepted Answer

First, $$z = \frac{85 - 100}{15} = -1. $$The probability is 0.1587.

Question 17

If a value is more than 3 standard deviations from the mean in a normal distribution, how is it classified?

Accepted Answer

Very unusual

Question 18

A company’s quality control department finds that the diameters of produced ball bearings are normally distributed with a mean of 10 mm and a standard deviation of 0.2 mm. What is the probability that a randomly selected ball bearing has a diameter between 9.8 mm and 10.2 mm? (Use the standard normal table: area to the left of $$z = 1 is 0.8413$$, area to the left of $$z = -1 is 0.1587.)$$

Accepted Answer

$$z_1$$ = \frac{9.8 - 10}{0.2} = -1$$, z_2$ = \frac{10.2 - 10}{0.2} = 1. $$Area between is $$0.8413 - 0.1587 = 0.6826.$$

Question 19

Which of the following is TRUE about the total area under any normal curve?

Accepted Answer

It is always equal to 1.

Normal Probability Distributions and the Standard Normal Distribution

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