6. Normal Distribution and Continuous Random Variables

Which of the following statements is true about a z-score?

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of about \$44,000 and a standard deviation of about \$2450. Use this information in Exercises 7–10.

Out of 800 residents, about how many would you expect to have a disposable income of between \$40,000 and \$42,000?

The towing capacities (in pounds) of all the pickup trucks at a dealership have a bell-shaped distribution, with a mean of 11,830 pounds and a standard deviation of 2370 pounds. In Exercises 45– 48, use the corresponding z-score to determine whether the towing capacity is unusual. Explain your reasoning.

5,500 pounds

The mean gestational length of a sample of 208 horses is 343.7 days, with a standard deviation of 10.4 days. The data set has a bell-shaped distribution.

b. Determine whether a gestational length of 318.4 days is unusual.

The mean annual salary for a sample of electrical engineers is \$86,500, with a standard deviation of \$1500. The data set has a bell-shaped distribution.

b. The salaries of three randomly selected electrical engineers are \$93,500, \$85,600, and \$82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.

Find the positive z-score for which 94% of the distribution’s area lies between -z and z.

Using a standard normal distribution, what is the $z$-score such that the area to the left of $z$ is $0.52$?

What braking distance represents the first quartile?

Government Solutions?In May, 2000, the Gallup Organization reported that 11% of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. Suppose a survey of a random sample of 1100 adult Americans finds that 84 have a great deal of trust and confidence in the federal government handling domestic issues. Would these results be considered unusual? Why?

In Exercises 33 and 34, find the indicated probabilities. If convenient, use technology to find the probabilities.

The daily surface concentration of carbonyl sulfide on the Indian Ocean is normally distributed, with a mean of 9.1 picomoles per liter and a standard deviation of 3.5 picomoles per liter. Find the probability that on a randomly selected day, the surface concentration of carbonyl sulfide on the Indian Ocean is

a. between 5.1 and 15.7 picomoles per liter.

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the z-score that has 78.5% of the distribution’s area to its left.

Pulse Rates. In Exercises 13–24, use the data in the table below for pulse rates of adult males and females (based on Data Set 1 “Body Data” in Appendix B). Hint: Draw a graph in each case.

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Find the probability that a male has a pulse rate between 70 beats per minute and 90 beats per minute.

In Exercises 5 and 6, 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. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (a) exactly 7. Identify any unusual events. Explain.

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

Water Footprint A water footprint is a measure of the appropriation of fresh water. The water footprint (in cubic meters) for a kilogram of wheat can be approximated by a normal distribution, as shown in the figure. (Source: Ecological Indicators)

b. What water footprint represents the 29th percentile?