Q: Find the margin of error for a 95% confidence interval when $$\sigma = 10.4$$ and $$n = 120.$$

$$E = 1.96 \times \frac{10.4}{\sqrt{120}} \approx 1.86$$

Q: A pollster wishes to estimate the proportion of United States voters who favor capital punishment. How large a sample is needed to be 90% confident that the sample proportion will not differ from the true proportion by more than 6%?

$$n = \left(\frac{1.645}{0.06}\$$right)^2$$ \times 0.25 \approx 188$$

Q: The standard IQ test has a mean of 103 and a standard deviation of 15. We want to be 98% certain that we are within 4 IQ points of the true mean. What is the required sample size?

$$n = \left(\frac{2.33 \times 15}{4}\$$right)^2$$ \approx 76$$

Question 1

Suppose the SAT scores are normally distributed with a mean of 1474 and a standard deviation of 295. What is the z-score for a student who scored 1930 on the SAT?

Accepted Answer

$$z = \frac{1930 - 1474}{295} = 1.55$$

Question 2

A variable is said to be normally distributed if its histogram is:

Accepted Answer

Symmetrical and bell-shaped

Question 3

The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy lasts less than 250 days?

Accepted Answer

$$P(X \u003c 250) = P\left(Z \u003c \frac{250 - 268}{15}\right) = P(Z \u003c -1.20) \approx 0.1151$$

Question 4

An airline knows from experience that the number of suitcases lost each week on a certain route is approximately normal with $$\mu = 15.5$$ and $$\sigma = 3.6. $$What is the probability that more than 20 suitcases are lost in a week?

Accepted Answer

$$P(X \u003e 20) = P\left(Z \u003e \frac{20 - 15.5}{3.6}\right) = P(Z \u003e 1.25) \approx 0.1056$$

Question 5

The distribution of cholesterol levels in teenage boys is approximately normal with $$\mu = 170$$ and $$\sigma = 30. $$What percent of teenage boys have cholesterol levels between 170 and 225?

Accepted Answer

$$P(170 \u003c X \u003c 225) = P(0 \u003c Z \u003c 1.83) \approx 0.4661$$

Question 6

The scores on a mathematics exam have a mean of 78 and a standard deviation of 8. What is the x-value that corresponds to a z-score of -1.28?

Accepted Answer

$$x = \mu + z\sigma = 78 + (-1.28)(8) = 67.76$$

Question 7

Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 100 women are randomly selected, what is the probability that their average height is greater than 63.0 inches?

Accepted Answer

$$P(\bar{X} \u003e 63.0) = P\left(Z \u003e \frac{63.0 - 63.6}{2.5/\sqrt{100}}\right) = P(Z \u003e -2.4) \approx 0.9918$$

Question 8

Find the margin of error for a 95% confidence interval when $$\sigma = 10.4$$ and $$n = 120.$$

Accepted Answer

$$E = 1.96 	imes \frac{10.4}{\sqrt{120}} \approx 1.86$$

Question 9

A random sample of 40 students has a mean annual earnings of $$3120. Assume the population standard deviation is 677. $$Construct the 95% confidence interval for the population mean $$\mu.$$

Accepted Answer

$$(2900, 3340)$$

Question 10

A sample of 10 randomly selected women has a mean height of 63.4 inches. The population standard deviation is 2.4 inches. Construct the 95% confidence interval for the population mean.

Accepted Answer

$$(61.9, 64.9)$$

Question 11

A survey of 100 fatal accidents showed that 23 were alcohol related. What is the point estimate for $$p$$, the population proportion of accidents that were alcohol related?

Accepted Answer

$$\hat{p} = \frac{23}{100} = 0.23$$

Question 12

When 415 college students were surveyed, 195 said they own their car. Construct a 95% confidence interval for the proportion of college students who say they own their cars.

Accepted Answer

$$(0.422, 0.518)$$

Question 13

A pollster wishes to estimate the proportion of United States voters who favor capital punishment. How large a sample is needed to be 90% confident that the sample proportion will not differ from the true proportion by more than 6%?

Accepted Answer

$$n = \left(\frac{1.645}{0.06}\$$right)^2$$ 	imes 0.25 \approx 188$$

Question 14

The standard IQ test has a mean of 103 and a standard deviation of 15. We want to be 98% certain that we are within 4 IQ points of the true mean. What is the required sample size?

Accepted Answer

$$n = \left(\frac{2.33 	imes 15}{4}\$$right)^2$$ \approx 76$$

Question 15

In a random sample of 28 families, the average weekly food expense was $$95.60 with a standard deviation of 22.50. $$Should a normal distribution or a t-distribution be used to construct a confidence interval for the mean?

Accepted Answer

t-distribution, because the sample size is less than 30 and the population standard deviation is unknown

Question 16

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

Accepted Answer

It is skewed to the right

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