Stat 152, Fall 2004

Final

SHOW YOUR WORK

NAME:

ID:

Q1
Q2
Q3
Q4
Q5
Q6
Q7
Extra
Total
Full Mark / 100+10
  1. (18 Points) Suppose you want to estimate the percentage of persons who have been immunized against polio in Gilbert (population 59,000) and can take a SRS of persons.

(a). (7 Points) What should be your sample size if you want the estimate to have margin of error of 4 percentage points with 95% confidence?

(b). (7 Points) An SRS of size 1000 was taken, and 257 persons in the sample were immunized against polio. Estimate the percentage of persons who have been immunized against polio in Gilbert, and give a 95% confidence interval for your estimation.

(c) (4 Points) If 100 SRS samples were taken and 100 95% confidence intervals were constructed for the percentage of persons who have been immunized against polio in Gilbert. What is the probability that one such interval will cover the true Gilbert immunization percentage?

  1. (20 pts) A population consists of 8 individuals. Each individual lives in one of the three cities: City A, City B or City C. These 8 individual’s heights are described by the table below:

1 / 2 / 3 / 4 / 5 / 6 / 7 / 8

Height

/ 70 / 60 / 60 / 55 / 60 / 50 / 65 / 70
City / A / A / A / B / B / B / C / C

A statistician, who knows nothing about the individuals, wants to find the average height of the population. The statistician considers the following two methods. Please provide an unbiasedestimate for the average population height and calculate the standard error of your estimate for each method.

a. (10 pts) Two cities are chosen at random without replacement and all the individuals live in the selected cities are included in sample. Assume Cities A and C are selected.

b. (10 pts) Two city clusters are chosen at random without replacement and two individuals in each selected city are randomly chosen and included in sample. Assume CityA and CityC are selected and individuals 1, 2, 7, 8 are in sample.

  1. (18 pts) A town has four supermarkets, ranging in size from 100 square meters (m2) to 1000 m2. We want to estimate the total amount of sales in the four stores for last month by sampling two of the stores without replacement. The sampling probability is proportional to the size of the store. Assume that store A and store D have been selected. Please estimate the total amount of sales using HT estimator. What is the standard error of your estimate?
  1. (6 pts) A stratified sample is being designed to estimate the prevalence p of a rare characteristic -- say, the proportion of residents in Milwaukee who have Lyme disease. Stratum 1, with N1 units, has a high prevalence of the characteristic; stratum 2, with N2 units, has low prevalence. Assume that the cost to sample a unit (for example, the cost to select a person for the sample and determine whether he or she has Lyme disease) is the same for each stratum and that at most 2000 units are to be sampled.Let p1 and p2 be the respective proportions in stratum 1 and stratum 2 with the rare characteristic. If p1=0.20, p2=0.02, and N1/N=0.4, what are n1 and n2 under optimal allocation?

  1. (22 pts) Investigators selected an SRS of 200 high school seniors from a population of 2000 for a survey of TV-viewing habits, with an overall response rate of 75%. By checking school records, they were able to find the grade point average (GPA) for the nonrespondents and classify the sample accordingly:

GPA / Sample Size / Number of Respondents / Hours of TV
3.00-4.00 / 75 / 70 / 30 / 15
2.00-2.99 / 70 / 55 / 40 / 20
Below 2.00 / 55 / 25 / 50 / 25
Total / 200 / 150
  1. (6 pts) What is the estimate for the average number of hours of TV watched per week if only respondents are analyzed?
  1. Additional problem (extra points 5 pts). What is the standard error of your above estimate?[Hints for standard error calculation:

1. ;

2. with ]

  1. (10 pts) Use the GPA classification to adjust the weights of the respondents in the sample. What is the weighting-class estimate of the average viewing time?
  1. (6 pts) The population counts are 700 students with a GPA between 3 and 4; 800 students with a GPA between 2 and 3; and 500 students with a GPA less than 2. Use these population counts to construct a poststratified estimate of the mean viewing time.
  1. (8 pts) There are 5 clusters in a population. Two clusters are selected at random without replacement and all the units in those two clusters are included in sample. Let be the number of units in each cluster. Let I(1) and I(2) be the indices of the first and second cluster selected. Find and .

  1. (8 pts) In a simple random sample of n out of N units, only m units respond. A follow-up survey was conducted on a simple random sample of k of the nonrespondents, and all of these k units responded in the follow-up.

For some characteristic of interest, if is the average response from the m units who originally responded, and is the average response in the follow-up, please provide an unbiased estimate for the population average ().

Additional problem (extra 5 pts). Prove that your above estimate is unbiased.Please make appropriate assumptions if you see a need.