Exam 1 FR 3218/5218 Semester II, 2005

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 = 53.

(6) 1. We commonly use systematic sampling to select sample units in our geographic-based natural resource populations.

a. Why are we more confident in systematically-selected samples (feel they give us better estimates of population parameters) in comparison to simple random samples of the same size?

b. Why is there a problem in computing a defensible estimate of precision for these systematic samples we have more confidence in?

(6) 2. To complete a sample size calculation for any sampling design we need two pieces of information (beyond knowing the size of the population). What are they?

1.

2.

(8) 3. Simple random sampling without replacement was used within one stratum of a population of interest. The sampling frame for the stratum lists 60 units. The observed values for the variable of interest on the selected sample units were:

20 23 28 21 30

a. Estimate the mean per unit in this one stratum.

b. What is the standard error of the estimate of mean per unit for this one stratum?

(10) 4. Simple random sampling without replacement was also used in the two additional strata for the population referred to in question 3 (question 3. considered stratum 1 only). The results were:

(a)	(b)	(c)	(d)	(e)
Stratum	Units in sample	Units in frame	Estimate of mean per unit	Standard error of (d)
1
2	10	10	100
3	3	30	10	1

a. Fill in the blanks in the table (including the standard error in stratum 2).

b. Estimate the population total of the variable of interest.

c. Find a 95% confidence interval for the population total of the variable of interest.

d. What would change in your analysis if I told you the stratum sizes had been estimated?

(6) 5. I am interested in estimating the total number of acres of Minnesota farmland that is expected to be in the Conservation Reserve Program for 2005 (some farmers can set aside some part of their farm for the program). I have a list of all farms in Minnesota so I can take a valid simple random sample of farms and query the farm owner for number of CRP acres for each farm in the sample. How might I improve upon that simple random sample by using each of the following approaches? That is, 1. what other information would I need, 2. how would I use that information, and 3. what assumptions would I be making? Note: there is not a single, correct answer for any approach.

a. stratified sampling

b. regression/ratio sampling (estimation)

c. unequal probability sampling (pps specifically)

(4) 6. In double (two-phase) sampling one takes a large and a small sample. We applied this idea to regression/ratio sampling (estimation) and stratification. Identify what was observed in each sample for these two cases and why it was observed?

a. double sampling for regression/ratio estimation

b. double sampling for stratification

(10) 7. A random sample without replacement of 5 units out of 50 possible was obtained for a population of interest. Both the variable of interest (Y) and an (much cheaper to measure) auxiliary variable (X) were observed on each selected unit. The true population total of X is known to be 2000. The sample values of Y and X were:

Unit	1	2	3	4	5
Y	100	120	90	150	130
X	28	48	30	55	50

Attached is an Excel regression analysis of these data that you should find useful. Here are some other summaries of the data, some of which may be of interest:

a. Estimate the population mean per unit for the variable of interest using a regression estimator.

b. Find the standard error of your estimate in a. (you only need to fill in the components of the proper formula for full credit)

c. Estimate the population mean per unit for the variable of interest using a ratio-of-means estimator.

d. What about the Excel output suggests favoring the estimator in a. or the estimator in c.? Which of the two estimators does it suggest favoring?

e. Does this appear to be a good application of regression/ratio sampling (estimation)? Why/why not?

(3) 8. You are told you will use a total of 100 sample units to estimate a quantity of interest for a large, heterogeneous population about which you know very little. However, there is obvious potential for applying stratification (i.e. four distinct strata are obvious) to improve the precision of your estimate. Why might you choose to use proportional allocation of the 100 units to strata rather than optimal allocation?