Math Chapter Seven Sample Exam

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1 Math Chapter Seven Sample Exam 1. The cereal boxes coming off an assembly line are each supposed to contain 12 ounces. It is reasonable to assume that the amount of cereal per box has an approximate normal distribution. A random sample of 16 boxes yields a mean of ounces and a standard deviation of 0.22 ounces. (d) Use the given data to find a 95% confidence interval for the mean amount of cereal per box. Based on the results in part, should we conclude that the assembly line equipment needs to be adjusted, and if so, how? What is the smallest sample size that could be used to estimate the mean amount of cereal per box with a 95% confidence interval so that the maximum error of the estimate is 0.03 ounces? Find a 95% confidence interval for the standard deviation of the amount of cereal per box on the assembly line.

2 2. Data is taken to compare the proportion of complaints at two airline terminals. In a random sample of 500 customers at the Transburg terminal, 51 have complaints. In a random sample of 400 customers at the Central City terminal, 48 have complaints. Use the given data to find a 90% confidence interval for the difference in proportion of complaints for the two terminals. Based on the results in part, should we conclude that the proportion of complaints is higher at one terminal than the other, and if so, which terminal has the higher proportion of complaints? 3. Suppose X 1, X 2,..., X n is a random sample from a distribution having p.d.f. f(x) = ( 1) / x if x > 1, where 1 <. Find the maximum likelihood estimator for (and don't forget to use the second derivative test to show that the likelihood function is maximized). Find the method of moments estimator for.

3 4. Data on rainfall is taken to compare a new cloud seeding technique and a standard cloud seeding technique. We are willing to assume that rainfall has an approximate normal distribution with the same variance for each technique. The mean and standard deviation for a random sample of 8 cloud seedings with the new technique are respectively 3.9 inches and 0.91 inches. The mean and standard deviation for a random sample of 12 cloud seedings with the standard technique are respectively 2.1 inches and 0.75 inches. Use the given data to find a 95% confidence interval for the difference in mean rainfall for the two cloud seeding techniques. Based on the results in part, should we conclude that one technique is better than the other, and if so, which one is better? Find a 95% confidence interval for the ratio of the variances of rainfall from the two techniques.

4 5. In a random sample of 1000 voters, 650 say that they will vote for Ms. Jones if she is a candidate in an upcoming election. Use the given data to find a 99% confidence interval for the true proportion of voters who will vote for Ms. Jones. Based on the results in part, should Ms. Jones conclude she will win the election? What is the smallest sample size that could be used to estimate the true proportion of voters who will vote for Ms. Jones with a 99% confidence interval so that the maximum error of the estimate is 0.01?

5 6. Suppose X 1, X 2,..., X n is a random sample from a distribution having p.d.f. f(x) = e x/(2 ) / (2 ) if x > 0. Prove that for each i, the moment generating function M(t) of X i / is the same as the moment generating function for a chi-square random variable (if t < 1/2), and state what the degrees of freedom is. Given the result in part, explain why it must be true that if Y = X 1 + X X n, then Y / has a chi-square distribution, and state what the degrees of freedom is. Suppose n = 6, and the following sample values are observed: , 3.399, , , 3.726, Use the result in part to derive a 95% confidence interval (based on Y) for the unknown parameter. (d) What must be done to shorten the length of the 95% confidence interval in part without lowering the confidence level?

6 7. Suppose X is a random variable having p.m.f. f(x) = if x = 1, 0, 1. Show that E(X) = 0. Find E(X 2 ), and state why X 2 is not an unbiased estimator of. Write a function of the statistic X 2 which is an unbiased estimator of. 8. Suppose X 1, X 2,..., X n is a random sample from a Poisson( ) distribution. Show that each of X ) and S 2 is an unbiased estimator of. Explain how you would decide which of the two estimators in part is better. (Note: you do not have to do anything except just explain.)

7 Here are some additional practice exercises from the textbook (with answers in the back of the textbook): 7.1-1, 7.1-5, 7.1-7, 7.1-9, , , 7.2-7, 7.3-9, , 7.4-1, , 7.5-7, , 7.6-1, , 7.7-5, , 7.9-3

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