STAT 3507 Midterm Review Problems I
1. A survey by Group Attitudes Inc., was said to measure American attitudes toward college. The polling firm mailed questionnaires to a probability sample of 4200 people with children across the United States and received 1188 responses. About 55% of those 1188 said they had major concerns about being to pay for their children's college education. Would you regard this figure as highly reliable and representative of the true proportion of American families (with children) who have this concern? Why?
2. A SRS of size 3 is taken from a population of size 5. In the population the random variable of interest Y takes the values: y = 2 1 3 1 3
a) Calculate the mean μ and variance σ2 of y for the population.
b) For this case, show numerically that and that is an unbiased estimator of σ2.
3. State park officials were interested in the proportion of campers who considered the campsite spacing adequate in a particular campground. They took a SRS of 30 from the first N = 300 camping parties that visited the campground on a particular weekend. Let yi = 0 if the ith party sampled does not think the campsite spacing is adequate and yi = 1 if it does (i = 1,2, ...30). If they found = 25, estimate with 90% confidence, p, the proportion of the 300 camping parties that found the campsite spacing adequate.
4. Use the data in #3 to determine the sample size that would have been needed to estimate p to within .05 with 95% confidence.
5. Resources managers of forest game lands are concerned about the size of the deer and rabbit populations during the winter months in a particular forest. As an estimate of the population sizes they propose using the average number of pellet groups for rabbits and deer per 30-foot-square plots. From an aerial photograph, the forest was divided into N = 10,000 30-foot-square grids. A simple random sample of 500 plots was taken and the number of pellet groups observed for rabbits and for deer giving the following results.
Deer sample mean = 2.30 sample variance = 0.65
Rabbits sample mean = 4.52 sample variance = 0.97
Estimate with 95% confidence, μ1 and μ2 the average number of pellet groups for deer and rabbits, respectively, per 30-square foot plots.
6. Ann Landers (1976) asked readers of her column to respond to this question: "If you had to do it over again, would you have children?" About 70% of the readers who responded said "NO". She received over 10,000 responses, 80% of them from women. If possible, give the target population, sampling frame, sampling unit and element. Discuss any possible sources of selection bias.
7. At one university there were 807 faculty members and research specialists in the College of Liberal Arts and Science in 1993; the list of faculty and their reported publications for 1992-1993 were available on the computer system. For each faculty member, the number of refereed publications was recorded. This number is not directly available on the database, so the investigator is required to examine each record separately. A frequency table for number of refeered publication is given for a SRS of 50 faculty members.
Refeered
publications 0 1 2 3 4 5 6 7 8 9 10
Faculty members 28 4 3 4 4 2 1 0 2 1 1
a) Plot the data using a histogram. Describe the shape of the data.
b) Estimate the mean number of publications per faculty member and find the standard error for you estimate (i.e. the square root of the variance).
c) Do you think that the sample mean from part (b) will be normally distributed? Why or why not?
d) Would you recommend giving a confidence interval for your estimate of the mean number of publications per faculty member? Why or why not?
e) Give a point estimate of the proportion of faculty members with no publications and give find the standard error of your estimate.
8. One quantity that is often of interest for a medical clinic is the percentage of patients that are overdue for a vaccination. Some clinics examine every record to determine that percentage; in a large practice, though, taking a census of the records can be time-consuming. Cullen (1994) took a sample of the 580 children served by an Auckland family practice to estimate the proportion of interest.
a) What sample size in a simple random sample would be necessary to estimate the proportion with 95% confidence and a margin of error of 0.10?
b) What sample size would be required for the same margin of error if the confidence desired was 90%?
9. What stratification variable(s) would you use for each of the following situations?
a) A telephone survey of students at your university to estimate the total amount of money students spend on textbooks.
b) A sample of public libraries in Ontario to study the availability of computer resources and the per capita expenditures.
c) A survey of anglers visiting a freshwater, to learn about which species of fish are preferred.
10. The following results were obtained from a stratified random sample:
Stratum 1: N1 = 100 n1 = 50
Stratum 2: N2 = 50 n2 = 50
Stratum 3: N3 = 300 n3 = 50
a) Estimate for the whole population
b) Give a 95% confidence interval for the population mean.
11. Allocate a total sample size of n = 100 between two strata having sizes N1 = 200 and N2 = 300 and variances and
a) using proportional allocation
b) using optimal allocation and assuming equal costs.
12. A botanical researcher wishes to design a survey to estimate the number of birch trees in a study area. The study area has been divided into 1000 units or plots. From a previous study, the variance in the number of trees per plot was estimated at 45. Using simple random sampling, what sample size size should be used to estimate the total number of trees in the study area
a) to within 500 trees with approximate 95% confidence?
b) to within 1000 trees with approximate 99% confidence?
13. A forester wants to estimate the total number of farm acres planted in trees for a state. Since the number of trees varies considerably with the size of the farm, he decides to stratify on farm size. The 240 farms in the state are placed in one of 4 categories according to size. A stratified random sample of 40 farms yields the results shown on number of acres planted in trees.
Stratum I0 – 200 acres / Stratum II
200 - 400 acres / Stratum III
400 - 600 acres / Stratum IV
over 600 acres
N1 = 86
n1 = 14 / N2 = 72
n2 = 12 / N3 = 52
n3 = 9 / N4 = 30
n4 = 5
97 67
42 125
25 92
105 86
27 43
45 59
53 21 / 125 155
67 96
256 47
310 236
220 352
142 190 / 142 256
310 440
495 510
320 396
196 / 167 655
220 540
780
where for stratum 1:
for stratum 2:
for stratum 3:
for stratum 4:
a) Estimate the total number of trees on farms in the state
b) Assuming the CLT for stratified random sampling applies (which may be a bit doubtful here), find a 90% confidence interval for the total number of trees on farms in the state.
c) Was proportional allocation used in taking this sample? Why or why not?