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Chapter 5: Q13E (page 314)

Will the sampling distribution ofχ¯ always be approximately normally distributed? Explain

Short Answer

Expert verified

A sampling distribution is statistics derived by continuous sampling from a greater populace.

Step by step solution

01

Sampling distribution

A sampling distributionis a probabilistic distribution of a statistic resulting from the selection of randomized samples from a particular population. It reflects the distribution of frequency on how far apart certain events will be for a specific demographic.

02

Explanation

No, since the central limit theorem asserts that if the sample size increases sufficient, the sampling distributions of x overbar are nearly distributed normally.

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Most popular questions from this chapter

Cable TV subscriptions and “cord cutters.” According to a recent Pew Research Center Survey (December 2015), 15% of U.S. adults admitted they are “cord cutters,” i.e., they canceled the cable/satellite TV service they once subscribed to. (See Exercise 2.4, p. 72) In a random sample of 500 U.S. adults, let pn represent the proportion who are “cord cutters.”

a. Find the mean of the sampling distribution of p^.

b. Find the standard deviation of the sampling distribution of p^.

c. What does the Central Limit Theorem say about the shape of the sampling distribution of p^?

d. Compute the probability that p^is less than .12.

e. Compute the probability that p^is greater than .10.

Improving SAT scores. Refer to the Chance(Winter2001) examination of Scholastic Assessment Test (SAT)scores of students who pay a private tutor to help them improve their results, Exercise 2.88 (p. 113). On the SAT—Mathematics test, these students had a mean score change of +19 points, with a standard deviation of 65 points. In a random sample of 100 students who pay a private tutor to help them improve their results, what is the likelihood that the sample mean score change is less than 10 points?

Rental car fleet evaluation. National Car Rental Systems, Inc., commissioned the U.S. Automobile Club (USAC) to conduct a survey of the general condition of the cars rented to the public by Hertz, Avis, National, and Budget Rent-a-Car.* USAC officials evaluate each company’s cars using a demerit point system. Each car starts with a perfect score of 0 points and incurs demerit points for each discrepancy noted by the inspectors. One measure of the overall condition of a company’s cars is the mean of all scores received by the company (i.e., the company’s fleet mean score). To estimate the fleet mean score of each rental car company, 10 major airports were randomly selected, and 10 cars from each company were randomly rented for inspection from each airport by USAC officials (i.e., a sample of size n = 100 cars from each company’s fleet was drawn and inspected).

a. Describe the sampling distribution of x, the mean score of a sample of n = 100 rental cars.

b. Interpret the mean of x in the context of this problem.

c. Assumeμ=30 and σ=60for one rental car company. For this company, findPx¯⩾45 .

d. Refer to part c. The company claims that their true fleet mean score “couldn’t possibly be as high as 30.” The sample mean score tabulated by USAC for this company was 45. Does this result tend to support or refute the claim? Explain.

Suppose a random sample of n = 25 measurements are selected from a population with mean μand standard deviation s. For each of the following values of μand role="math" localid="1651468116840" σ, give the values of μχ¯ and σχ¯.

  1. μ=100,σ=3
  2. μ=100,σ=25
  3. μ=20,σ=40
  4. μ=10,σ=100

Analysis of supplier lead time. Lead timeis the time betweena retailer placing an order and having the productavailable to satisfy customer demand. It includes time for placing the order, receiving the shipment from the supplier, inspecting the units received, and placing them in inventory. Interested in average lead time,, for a particular supplier of men’s apparel, the purchasing department of a national department store chain randomly sampled 50 of the supplier’s lead times and found= 44 days.

  1. Describe the shape of the sampling distribution ofx¯.
  2. If μand σare really 40 and 12, respectively, what is the probability that a second random sample of size 50 would yieldx¯ greater than or equal to 44?
  3. Using the values forμ and σin part b, what is the probability that a sample of size 50 would yield a sample mean within the interval μ±2σn?
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