STAT 3507 Exercise Questions Solutions

STAT 3507 Exercise Questions Solutions

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Bernoulli Sampling Questions 

Cluster Sampling Questions

Ratio, Regression and Difference Estimation

Stratified Random Sampling

Systematic Sampling

Two-stage Cluster Sampling Questions

1-A university professor who is correcting 600 written examinations decides to get a preliminary idea of the passing rate on the test. He decides to use a simple randomized scheme to single out a smaller number of exam copies for first-hand correction. In passing through the pile of exams, he tosses an ordinary six-sided die, once for each exam copy. If the die shows a 6, he corrects the corresponding exam, otherwise not. Suppose the sample selected in this way consists of 90 students and that 60 out of these are found to have passed.

a.               Identify the sampling design implemented.

b.                 Estimate the total number of students who passed the exams

c.               Compute a 95% confidence interval, based on the normal approximation, for the number of passing students (among the 600).

d.              Repeat (b) and (c) using the alternative (improved) estimator.

2-

Table: Number of Employees and Range

Division	Number of employees	Cumulative range
1	1200	1— 1200
2	450	1201 —1650
3	2100	1651— 3750
4	860	3751— 4610
5	2840	4611 — 7450
6	1910	7451— 9350
7	390	9361 — 9750
8	3200	9751 - 12950
	12950

i.                        Suppose 2011, 7972 and 10281 are the random numbers generated between 1 and 12950. Using this information what are the clusters selected?

ii.                      Suppose the total number of sick days used by the three sampled divisions during the past quarter are respectively, y, = 4320 y₂ = 4160 y₃ = 5790 .

Estimate the average number of sick clays used per person for the entire firm and place a bound on the error of estimation.

Ratio and Regression Estimators (Examples) under Simple Random Sampling Without Replacement

Set 1: Estimation of Ratio

Let us consider a community having eight community areas. Suppose that we wish to estimate the ratio R of total pharmaceutical expenses Y to total medical expenses X among all persons in the community. To do this, a simple random sample of two community areas is to be taken and every household in each sample community area is to be interviewed. The data for the community areas am as given in Table 1 below:

Table 1: Pharmaceutical Expenses and Total Medical Expenses Among All