6. Normal Distribution and Continuous Random Variables

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x > 65)

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 0.89 < z < 0)

Standard Normal Distribution. In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Birth Weights Based on Data Set 6 “Births” in Appendix B, birth weights of girls are normally distributed with a mean of 3037.1 g and a standard deviation of 706.3 g.

c. What is the value of the mode?

In Section 10.2, we tested hypotheses regarding a population proportion using a z-test. However, we can also use the chi-square goodness-of-fit test to test hypotheses with k = 2 possible outcomes. In Problems 25 and 26, we test hypotheses with the use of both methods.

Living Alone? In 2000, 25.8% of Americans 15 years of age or older lived alone, according to the Census Bureau. A sociologist, who believes that this percentage is greater today, conducts a random sample of 400 Americans 15 years of age or older and finds that 164 are living alone.

a. If the proportion of Americans aged 15 years or older living alone is 0.258, compute the following expected numbers: Americans 15 years of age or older who live alone; Americans 15 years of age or older who do not live alone.

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=20, p=0.65, q=0.35

Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B).

b. If 9 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg.

In Exercises 69 and 70, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.

A survey of U.S. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (b) exactly 50.

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.

z = -2.37

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.

d. Is the new capacity of 20 passengers safe?

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x≥109)

a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)

The random variable x is normally distributed with the given parameters. Find each probability.

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)

c. What weight represents the first quartile?

Heights On the basis of Data Set 1 “Body Data” in Appendix B, assume that heights of men are normally distributed, with a mean of 68.6 in. and a standard deviation of 2.8 in.

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a. The U.S. Coast Guard requires that men must have a height between 60 in. and 80 in. Find the percentage of men who satisfy that height requirement.