Q: If the mean score on a standardized test is 500 and the standard deviation is 100, what is the x-value that is 1.5 standard deviations above the mean?

$$x = 500 + 1.5 \times 100 = 650$$

Q: A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed with a mean of 9 pounds and a standard deviation of 2 pounds. What is the weight corresponding to a z-score of -0.44?

$$x = 9 + (-0.44) \times 2 = 8.12$$ pounds

Question 1

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

Accepted Answer

The total area under the curve is less than 1.

Question 2

A college student works 5 hours per week. The mean number of hours worked by college students with jobs is 25 hours, with a standard deviation of 11 hours. What is the z-score for this student?

Accepted Answer

$$z = \frac{5 - 25}{11} = -1.82$$

Question 3

Using the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 4

If the mean score on a standardized test is 500 and the standard deviation is 100, what is the x-value that is 1.5 standard deviations above the mean?

Accepted Answer

$$x = 500 + 1.5 	imes 100 = 650$$

Question 5

A sample of 50 drivers aged 16-19 has a mean distance driven of 22.5 miles. The population mean is 20.7 miles and the population standard deviation is 6.5 miles. What is the z-score for this sample mean?

Accepted Answer

$$z = \frac{22.5 - 20.7}{6.5 / \sqrt{50}} = 1.96$$

Question 6

Which formula is used to convert an x-value to a z-score in a normal distribution?

Accepted Answer

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

Question 7

A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed with a mean of 9 pounds and a standard deviation of 2 pounds. What is the weight corresponding to a z-score of -0.44?

Accepted Answer

$$x = 9 + (-0.44) 	imes 2 = 8.12$$ pounds

Question 8

Which statement best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 9

For a population with mean $$\mu = 280$$ and standard deviation $$\sigma = 78$$, what is the probability that a randomly selected state has between 300 and 350 physicians per 100,000 residents? (Assume normality and use the z-score method.)

Accepted Answer

$$P(300 \u003c x \u003c 350) = P(z \u003c 0.897) - P(z \u003c 0.256) = 0.816 - 0.603 = 0.213$$

Question 10

A survey indicates that the time spent in a supermarket is normally distributed with a mean of 41 minutes and a standard deviation of 12 minutes. What is the probability that a shopper spends between 20 and 50 minutes in the store?

Accepted Answer

$$P(20 \u003c x \u003c 50) = P(z \u003c 0.75) - P(z \u003c -1.75) = 0.7734 - 0.0401 = 0.7333$$

Question 11

If the mean room and board expense at four-year colleges is $$11,806 with a standard deviation of 1$$,650, what is the probability that the mean expense for a sample of 9 colleges is less than $12,250?

Accepted Answer

$$z = \frac{12,250 - 11,806}{1,650 / \sqrt{9}} = 0.81$$, so $$P(x \u003c 12,250) = 0.7910$$

Question 12

Which of the following is the formula for the standard error of the mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Question 13

A binomial experiment has $$n = 45$$ trials and $$p = 0.40. $$Can the normal approximation be used, and what is the mean and standard deviation?

Accepted Answer

Yes, because $$np = 18$$ and $$nq = 27$$ are both greater than 5. Mean: $$\mu = np = 18$$, SD: $$\sigma = \sqrt{npq} = \sqrt{45 	imes 0.4 	imes 0.6} \approx 3.2$$

Question 14

When using the normal approximation for a binomial distribution, what is the purpose of the continuity correction?

Accepted Answer

To adjust for the fact that the binomial distribution is discrete and the normal distribution is continuous.

Question 15

In a study, 52% of drivers who use alcohol admit to driving aggressively. If you randomly select 200 drivers, what is the probability that at least 100 will say yes? (Use normal approximation and continuity correction.)

Accepted Answer

$$P(x \u003e 99.5) = P(z \u003e -0.64) = 0.7389$$

Question 16

A study of NFL retirees found that 12.3% have undergone knee joint replacement surgery. If you randomly select 100 former NFL players, what is the probability that exactly 15 have had the surgery? (Use normal approximation and continuity correction.)

Accepted Answer

$$P(14.5 \u003c x \u003c 15.5)$$, $$z_{14.5} = \frac{14.5 - 12.3}{\sqrt{100 	imes 0.123 	imes 0.877}} = 0.67$$, $$z_{15.5} = 0.98$$, so $$P = 0.8340 - 0.7486 = 0.0854$$

Question 17

If the cumulative area to the right of a z-score is 0.1075, what is the cumulative area to the left?

Accepted Answer

0.8925

Normal Probability Distributions and the Central Limit Theorem Chapter. 5

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