Q: Suppose the waiting time at a coffee shop is uniformly distributed between 0 and 10 minutes. What is the probability that a randomly selected customer waits more than 7 minutes?

$$P(X \u003e 7) = (10-7) \times \frac{1}{10-0} = 3 \times 0.1 = 0.3$$

Q: Given a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what value of $$X$$ corresponds to the 90th percentile?

$$X = 100 + 1.28 \times 15 = 119.2$$

Question 1

Which of the following statements best describes a density curve?

Accepted Answer

A density curve is a smooth curve that represents the distribution of a continuous variable, where the total area under the curve is 1.

Question 2

Suppose the waiting time at a coffee shop is uniformly distributed between 0 and 10 minutes. What is the probability that a randomly selected customer waits more than 7 minutes?

Accepted Answer

$$P(X \u003e 7) = (10-7) 	imes \frac{1}{10-0} = 3 	imes 0.1 = 0.3$$

Question 3

A variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the $$z-$$score for $$X = 65$$?

Accepted Answer

$$z = \frac{65-50}{10} = 1.5$$

Question 4

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

Accepted Answer

It has a mean of 1.

Question 5

The weights of three-year-old children are normally distributed with $$\mu = 30.7$$ pounds and $$\sigma = 3.6$$ pounds. What is the probability that a randomly selected child weighs at least 30 pounds?

Accepted Answer

$$P(X \geq 30) = 0.577$$

Question 6

Given a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what value of $$X$$ corresponds to the 90th percentile?

Accepted Answer

$$X = 100 + 1.28 	imes 15 = 119.2$$

Question 7

If the population standard deviation is $$\sigma = 12$$ and the sample size is $$n = 36$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = 2$$

Question 8

A sample of 100 college students is taken to estimate the proportion who work full-time. If 40 students in the sample work full-time, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{40}{100} = 0.4$$

Question 9

Which condition must be met for the sampling distribution of the sample proportion to be approximately normal?

Accepted Answer

$$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 10

The Central Limit Theorem states that as the sample size $$n$$ increases, the sampling distribution of the sample mean:

Accepted Answer

Approaches a normal distribution, regardless of the shape of the population distribution.

Question 11

A water taxi is rated for a load limit of 3500 lb and can carry 25 passengers. If the weights of men are normally distributed with $$\mu = 182.9 lb $$and $$\sigma = 40.8 lb$$, what is the probability that the average weight of 25 randomly selected men exceeds 140 lb?

Accepted Answer

$$P(\bar{X} \u003e 140) = 1$$

Question 12

If $$P(Z \u003c z) = 0.8461$$ for a standard normal variable, what is the value of $$z$$?

Accepted Answer

$$z = 1.02$$

Question 13

Which formula gives the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$

Question 14

A normal distribution has mean $$\mu = 45$$ and standard deviation $$\sigma = 8. $$What is the $$z-$$score for $$X = 36$$?

Accepted Answer

$$z = \frac{36-45}{8} = -1.125$$

Question 15

If the area to the right of a $$z-$$score is $$\alpha = 0.05$$, what is the critical value $$z_{0.05}$$?

Accepted Answer

$$z_{0.05} = 1.645$$

Question 16

A sample of 36 students has an average SAT score of 1600. If SAT scores are normally distributed with $$\mu = 1511$$ and $$\sigma = 312$$, what is the probability that the sample mean is higher than 1600?

Accepted Answer

$$P(\bar{X} \u003e 1600) = 0.043$$

Question 17

Which of the following best describes the effect of increasing sample size on the sampling error when estimating a population mean?

Accepted Answer

The sampling error tends to decrease as the sample size increases.

CH. 6 Normal Probability Distributions and Sampling Distributions: Study Guide

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