Question 7 A population is formed of 5 clusters. The cluster sizes Mi and the cluster totals Ti of some variable of interest are given in the table below. Cluster, i | 1 | 2 | 3 | 4 | 5 Total, Ti | 60 | 60 | 30 | 20 | 10 Size, Mi | 20 | 15 | 8 | 5 | 2 The cluster sizes are known, but the totals are not. A single cluster is sampled by simple random sampling and the total t and size m are observed for the sampled cluster. Two possible estimators of the population mean of the quantity of interest are t/m and t/Mave, where Mave is the average size of the 5 clusters. (a) Using the information in the table on the population, compute the expectation and variance of each of these estimators under simple random sampling of clusters. [8] (b) [Type] Comment on the results in (a): compare and explain them in this particular example, and comment on the implications for the general case of estimating a population mean from cluster sampling with variable cluster sizes. Word limit: 300 words. [9]
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For the estimator t/m: - Expectation: E(t/m) = E(t)/E(m) Since the cluster sizes are known, we can calculate the expectation of t as the sum of the cluster totals divided by the number of clusters: E(t) = (60 + 60 + 30 + 20 + M)/5 E(m) = Mave = (60 + 60 + Show more…
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n = 36 Sample mean (Xbar) = 20.76 S = 12 H0: µ ≤ 20 Ha: µ > 20 The standardized t test statistic equals A. 0.38 B. 4.6 C. -0.38 D. -2.3 Exhibit 7 (Questions 45-48) The following sample data are from a normal population: 2, 4, 3, 2, 4. Refer to Exhibit 7. What is the point estimate of the population mean? A. 3 B. 2.5 C. 1 D. 1.5 Refer to Exhibit 7. What is the point estimate of the population standard deviation? A. 1 B. 2.5 C. 1.5 D. 3 Refer to Exhibit 7. With 95% confidence, what is the margin of error for the estimation of the population mean? A. 0.953 B. 0.901 C. 1.241 D. 0.980 Refer to Exhibit 7. What is the 95% confidence interval for the population mean? A. 0.901 to 1.901 B. 1.759 to 4.241 C. 1.047 to 2.953 D. 0.020 to 1.980
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Question 6: In estimating a population mean with a confidence interval based on a random sample X1,...,Xn from a Normal distribution with an unknown mean and a known variance, how the length of the confidence interval changes if we increase the sample size? Question 7: A University of Florida economist conducted a study of Virginia elementary school lunch menus. During the state-mandated testing period, school lunches average 863 calories. The economist claimed that after the testing period end, the average calories content of Virginia school lunches dropped significantly. Set up the null and alternative hypothesis to test the economist's claim. Let m be the average calories content of Virginia school lunches. Question 8: Suppose you are interested in conducting the statistical test of H0 : mu = 200 against Ha: mu > 200, where mu is the mean of the population. Suppose you have decided to use the following decision rule: Reject H0 if the sample mean of a random sample of 100 items is more than 215. Assume that the standard deviation of the population is 80. Find the probability of making a Type I error, by using this decision rule.
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