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Chapter 7: Q.68 (page 456)

The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The central limit theorem says that

(a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean μ for all bulbs of this type.

(b) the average burnout time of a large number of bulbs has a distribution of the same shape (strongly skewed) as the population distribution.

(c) the average burnout time of a large number of bulbs has a distribution with a similar shape but not as extreme (skewed, but not as strongly) as the population distribution.

(d) the average burnout time of a large number of bulbs has a distribution that is close to Normal.

(e) the average burnout time of a large number of bulbs has a distribution that is exactly Normal.

Short Answer

Expert verified

The correct answer is (d).

Step by step solution

01

Given Information

The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right.

02

Explanation

According to the central limit theorem, if the sample size of a sampling distribution is 30 or greater, the sample mean is nearly normal, with a mean of μ and a standard deviation of σn.

As a result, the average burnout time of a large number of bulbs has a Normal distribution.

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Most popular questions from this chapter

Doing homework Refer to Exercise 9.

(a) Make a graph of the population distribution given that there are 3000 students in the school. (Hint: What type of variable is being measured?)

(b) Sketch a possible graph of the distribution of sample data for the SRS of size 100 taken by the AP Statistics students.

The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5,but some traps have several moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation0.7.

(a) What are the mean and standard deviation of the average number of moths xin50traps?

(b) Use the central limit theorem to help you find the probability that the average number of moths in 50traps is greater than0.6.

The candy machine Suppose a large candy machine has 15% orange candies. Use Figure 7.13 (page 435) to help answer the following questions.

(a) Would you be surprised if a sample of 25 candies from the machine contained 8 orange candies (that's 32%orange)? How about 5 orange candies (orange)? Explain.

(b) Which is more surprising: getting a sample of 25 candies in which localid="1649737645698" 32%are orange or getting a sample of 50 candies in which localid="1649737650189" 32%are orange? Explain.

Suppose we select an SRS of size n=100from a large population having proportion p of successes. Let p∧ be the proportion of successes in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of p∧?

(a) 0.01

(b) 111

(c) 0.85

(d) 0.975

(e)0.999

Airline passengers get heavier In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) in 2003told airlines to assume that passengers average 190pounds in the summer, including clothes and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 30passengers.

(a) Explain why you cannot calculate the probability that a randomly selected passenger weighs more than 200pounds.

(b) Find the probability that the total weight of the passengers on a full flight exceeds 6000pounds. Show your work. (Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)
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