Exam 1 FR 3218/5218 Semester II, 2004
If it is not clear what a question is asking, request clarification from the instructor. Misreading a question is not grounds for partial credit. To receive partial credit for the calculation problems, formulas and intermediate calculations must be legibly shown. A good strategy would be to go through and answer/set up as much of each calculation question as you can and then go back to fill in details. It is my experience that divine intervention seldom occurs during examinations.
The number preceding the question number is the point value of that particular question. Total points = 34.
(3) 1. To determine the appropriate number of sample units to include in a sample (for any sample design) requires quantifying precision requirements. How does quantifying precision requirements relate to a statement of objective or study of the decision that underlies the sampling effort? Provide a concrete example.
(8) 2. The following observations were collected by a simple random sampling without replacement of three (3) elements out of a population of 30 elements total. Derive a 90% confidence interval for the population total.
Sample observations (yi’s): 2, 5, 8
Summaries
(use as you choose): 
Was this a very successful sampling effort?
(2) 3. We use coefficient of variation (Cv) to quantify variability among elements of a population. For simple random sampling with replacement, if coefficient of variation doubles, the number of sample units required to achieve a specified level of precision:
a. doubles
b. halves
c. increases by a factor of four
d. decreases by one fourth
(4) 4. PPS sampling was used to select one unit from the following population. The random number used to select the unit was 8. The observation on the variable of interest for the unit selected was 21.
|
Element |
Size |
|
Element |
Size |
|
|
1 |
1 |
|
4 |
3 |
|
|
2 |
2 |
|
5 |
1 |
|
|
3 |
3 |
|
6 |
2 |
|
What is the PPS estimate of the population total?
Had the same unit been selected by simple random sampling, what would the estimate of population total have been?
(4) 5. Given the following strata summaries, provide the stratified random sampling estimate of population mean per unit.
|
Stratum |
nh |
Nh |
|
Stratum |
nh |
Nh |
|
|
1 |
15 |
213 |
5 |
3 |
10 |
98 |
16 |
|
2 |
20 |
150 |
1 |
4 |
25 |
432 |
9 |
(3) 6. You can afford to take 100 units total from a population that has been stratified into two strata. The stratum relative sizes and estimates of variances among observations within strata are as follows:
|
Stratum |
Relative Size |
Variance Within Stratum |
|
1 |
0.3 |
1600 |
|
2 |
0.7 |
900 |
Will proportional or optimal (circle one) allocation place more of the 100 units in stratum 1? Assume sampling costs are the same for the two strata.
(3) 7. Interest lies in the amount of money spent by University of Minnesota students on, or because of, Valentine’s Day. Total number of students at the University is known and a complete list of students with campus contact information is available. It is felt that whether a student has a “special friend” or not likely impacts the amount that a student spends on, or because of, Valentine’s Day. Given this information, and only this information, what sampling design would you propose be used to select a sample of students to estimate the amount of money spent by University of Minnesota students on, or because of, Valentine’s Day? Justify your choice.
(3) 8. Data on a variable of interest and a related variable (whose true mean is known for the population) were collected via simple random sampling. You have been asked to determine whether a regression or ratio-of-means estimator could/should be used to estimate the population total.
What should you first do with the data to address the question of whether a regression or ratio-of-means estimator could/should be used to estimate the population total? What would you be looking for (I am looking for three things in particular)?
(4) 9. In the past, your employer has used simple random sampling to monitor the mean value of a critical environmental indicator on a weekly basis. A simple random sampling sample size of 10 units is required to meet the regulation standard. One observation of the environmental indicator costs $800. Your science team has identified a second variable that is highly correlated with the environmental indicator (correlation estimated to be 0.9) but is much cheaper to observe than the environmental indicator ($200 per unit).
What double sampling with a regression estimator sample sizes would you suggest be used in this situation? How much money would this save your employer over a year (52 weeks in a year)?
To answer the question you may use the graphs below, marking the relevant points you utilize. Or, if you wish, use the equations the graphs are based upon.
|
|
|
Sample sizes:
Cost savings: