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Chapter 6: Q110SE (page 330)
Calculate the finite population correction factor for each
of the following situations:
a. n = 50, N = 2,000
b. n = 20, N = 100
c. n = 300, N = 1,500
a.0.9876
b. 0.8989
c. 0.8947
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Question: Explain the differences in the sampling distributions of for large and small samples under the following assumptions.
a. The variable of interest, x, is normally distributed.
b. Nothing is known about the distribution of the variable x.
Largest private companies. IPOs—initial public offerings of stock—create billions of dollars of new wealth for owners, managers, and employees of companies that were previously privately owned. Nevertheless, hundreds of large and thousands of small companies remain privately owned. The revenues of a random sample of 15 firms from Forbes 216 Largest Private Companies list are given in the table below
a. Describe the population from which the random sample was drawn.
b. Use a 98% confidence interval to estimate the mean revenue of the population of companies in question
c. Interpret your confidence interval in the context of the problem
d. What characteristic must the population possess to ensure the appropriateness of the estimation procedure used in part b?
e. Suppose Forbes reports that the true mean revenue of the 216 companies on the list is $5.0 billion. Is the claim believable?
Scallops, sampling, and the law. Interfaces (March–April 1995) presented the case of a ship that fishes for scallops off the coast of New England. In order to protect baby scallops from being harvested, the U.S. Fisheries and Wildlife Service requires that “the average meat per scallop weigh at least 136 of a pound.” The ship was accused of violating this weight standard. Author Arnold Barnett lays out the scenario:
The vessel arrived at a Massachusetts port with 11,000 bags of scallops, from which the harbormaster randomly selected 18 bags for weighing. From each such bag, his agents took a large scoopful of scallops; then, to estimate the bag’s average meat per scallop, they divided the total weight of meat in the scoopful by the number of scallops it contained. Based on the 18 [numbers] thus generated, the harbormaster estimated that each of the ship’s scallops possessed an average of 139 of a pound of meat (that is, they were about seven percent lighter than the minimum requirement). Viewing this outcome as conclusive evidence that the weight standard had been violated, federal authorities at once confiscated 95 percent of the catch (which they then sold at auction). The fishing voyage was thus transformed into a financial catastrophe for its participants. The actual scallop weight measurements for each of the 18 sampled bags are listed in the table below. For ease of exposition, Barnett expressed each number as a multiple of of a pound, the minimum permissible average weight per scallop. Consequently, numbers below 1 indicate individual bags that do not meet the standard. The ship’s owner filed a lawsuit against the federal government, declaring that his vessel had fully complied with the weight standard. A Boston law firm was hired to represent the owner in legal proceedings, and Barnett was retained by the firm to provide statistical litigation support and, if necessary, expert witness testimony.
0.93 | 0.88 | 0.85 | 0.91 | 0.91 | 0.84 | 0.90 | 0.98 | 0.88 |
0.89 | 0.98 | 0.87 | 0.91 | 0.92 | 0.99 | 1.14 | 1.06 | 0.93 |
A random sample of 225 measurements is selected from a population, and the sample means and standard deviation are x = 32.5 and s = 30.0, respectively.
a. Use a 99% confidence interval to estimate the mean of the population, .
b. How large a sample would be needed to estimate m to within .5 with 99% confidence?
c. Use a 99% confidence interval to estimate the population variance, .
d. What is meant by the phrase99% confidence as it is used in this exercise?
A random sample of 70 observations from a normally distributed population possesses a sample mean equal to 26.2 and a sample standard deviation equal to 4.1.
a. Find an approximate 95% confidence interval for
b. What do you mean when you say that a confidence coefficient is .95?
c. Find an approximate 99% confidence interval for
d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?
e. Would your confidence intervals of parts a and c be valid if the distribution of the original population was not normal? Explain
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