Q: If 200 shoppers enter the store, and the probability that a shopper spends more than 39 minutes is 0.6915, how many shoppers would you expect to spend more than 39 minutes in the store?

$$200 \times 0.6915 = 138.3$$, so about 138 shoppers

Question 1

Which of the following best describes the shape and properties of a normal distribution?

Accepted Answer

It is bell-shaped, symmetric about the mean, and the total area under the curve is 1.

Question 2

Given a normal distribution with mean $$\mu = 45$$ and standard deviation $$\sigma = 12$$, what is the z-score for $$x = 54$$?

Accepted Answer

$$z = \frac{54 - 45}{12} = 0.75$$

Question 3

If the area to the left of $$z = 1.15 is 0.8749$$, what is the probability that a randomly selected value from the standard normal distribution is greater than 1.15?

Accepted Answer

$$P(z \u003e 1.15) = 1 - 0.8749 = 0.1251$$

Question 4

A survey indicates that people use their cellular phones an average of 1.5 years before buying a new one, with a standard deviation of 0.25 years. What is the probability that a randomly selected user will use their phone for less than 1 year? (Use the standard normal table.)

Accepted Answer

First, $$z = \frac{1 - 1.5}{0.25} = -2. $$The probability is $$P(z \u003c -2) = 0.0228$$

Question 5

For a standard normal distribution, what is the area between $$z = -1.5$$ and $$z = 1.25$$?

Accepted Answer

$$0.8944 - 0.0668 = 0.8276$$

Question 6

A job exam is normally distributed with a mean of 50 and a standard deviation of 10. What is the minimum score needed to be in the top 10% of applicants?

Accepted Answer

Find the z-score for the 90th percentile ($$z = 1.28$$), then $$x = 50 + 1.28 	imes 10 = 62.8$$

Question 7

Which formula is used to convert a z-score to a data value $$x in a $$normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 8

A population has $$\mu = 63$$ and $$\sigma = 11. If $$you take a random sample of $$n = 36$$, what is the standard deviation of the sample mean (standard error)?

Accepted Answer

$$\frac{11}{\sqrt{36}} = 1.833$$

Question 9

According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size $$n$$ increases?

Accepted Answer

It becomes approximately normal, regardless of the shape of the population distribution, as $$n$$ becomes large (typically $$n \geq 30).$$

Question 10

A population has $$\mu = 3173$$ and $$\sigma = 1120. $$What is the probability that a randomly selected undergraduate has a credit card balance less than $$2700$$? (Find the z-score and use the standard normal table.)

Accepted Answer

First, $$z = \frac{2700 - 3173}{1120} = -0.4232. $$The probability is $$P(z \u003c -0.42) \approx 0.3372$$

Question 11

If the area to the left of $$z = -0.99 is 0.1611$$, what is the probability that a value from the standard normal distribution is greater than $$z = -0.99$$?

Accepted Answer

$$1 - 0.1611 = 0.8389$$

Question 12

A supermarket shopper spends an average of 45 minutes in the store with a standard deviation of 12 minutes. What is the probability that a randomly selected shopper spends more than 39 minutes in the store?

Accepted Answer

First, $$z = \frac{39 - 45}{12} = -0.5$$. $$P(z \u003e -0.5) = 1 - 0.3085 = 0.6915$$

Question 13

If 200 shoppers enter the store, and the probability that a shopper spends more than 39 minutes is 0.6915, how many shoppers would you expect to spend more than 39 minutes in the store?

Accepted Answer

$$200 	imes 0.6915 = 138.3$$, so about 138 shoppers

Question 14

Which of the following statements about percentiles in a normal distribution is correct?

Accepted Answer

The 60th percentile is the value below which 60% of the data fall.

Question 15

A normal distribution has $$\mu = 70$$ and $$\sigma = 15. $$What is the value corresponding to the 87th percentile? (Use $$z \approx 1.13$$ for the 87th percentile.)

Accepted Answer

$$x = 70 + 1.13 	imes 15 = 86.95$$

Question 16

Which of the following is the correct formula for the standard error of the mean for a sample of size $$n$$ from a population with standard deviation $$\sigma$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Normal Probability Distributions and the Central Limit Theorem: Study Notes

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