Stat 152, Fall 2005 Midterm II SHOW YOUR WORK NAME: ID: Extra. Total. Full Mark 60+5

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1 Stat 152, Fall 2005 Midterm II SHOW YOUR WORK NAME: ID: Q1 Q2 Q3 Q4 Extra Total Full Mark 60+5

2 1. (24 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: Height 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 read the following methods carefully and provide an unbiased estimate for the average population height and calculate the standard error of your estimate for each method. a. (12 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. b. (12 pts) Two cities are chosen with un-equal probabilities with replacement. The selection probability is proportional to the city size, and all the individuals live in the selected cities are included in sample. Assume Cities A and C are selected.

3 2. (14 pts) A town has four supermarkets, ranging in size from 100 square meters (m 2 ) to 1000 m 2. 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. What is the standard error of your estimate?

4 3. (16 pts) Two lines are chosen at random from the below region. We observe the objects (ponds) that intersect the lines and record those ponds pollution level. The width of the ponds in that region is known: the width of pond A = 1.4 (miles); the width of pond B = 3 (miles); the width of pond C = 1.7 (miles); the width of pond D = 2.2 (miles); the width of pond E = 1.8 (miles); the width of pond F = 1.5 (miles). a. (4 pts) What is the probability of pond B in sample? b. (6pts) What is the probability that the sample includes and only includes ponds B, E and F? (hint: you should consider the width of overlapping region of the ponds) c. (6pts) Continuing with (b). If the pollutant concentrations (in parts per million) for the three ponds in the sample were 10, 5, and 2. Please give an unbiased estimate of the mean pollution concentration per pond in the population. You need NOT calculate the standard error of your estimate.

5 4. (6pts) Mitofsky-Waksberg method in telephone surveys (1). Construct a frame of all area codes and prefixes in the area of interest. (2). Draw a random sample of ten-digit telephone numbers from the set of telephone numbers with area code and prefix in the frame and suffix between 0000 and (3). Dial each number selected in step 2. If the selected number is residential, interview the household and choose its psu to be in the sample; the associated psu is the block of 100 telephone numbers that have the same first eight digits as the selected numbers. For example, if the randomly selected telephone number (202) is residential, then the psu of all numbers of the form (202)456-14XX is included in the sample. If the selected number is not residential, discard it and its psu. Continue sampling at the first stage until the desired number of psu s, n, is selected. (4). For the second stage of sampling, randomly select additional telephone numbers with the replacement from each psu in the sample until the desired sample size for each psu is attained. The above method dramatically increases the percentage of calls made that reach residential households, compared to Random Digit Dialing method, which randomly selects an area code and prefix combination known to be in use and appending a four-digit chosen randomly from 0000 to (a) (2 pts) In this sampling design (Mitofsky-Waksberg method), are the psu s selected with equal probabilities or un-equal probabilities? (b) (4 pts) Please argue briefly why Mitofsky-Waksberg method increases the percentage of calls made that reach residential households (compared to Random Digit Dialing method)?

6 Additional problem (for extra 5 pts). Lahiri s Method: Pairs of numbers are generated to select psu s and some of them are rejected if the psu size is too small. Let N = number of psu s in population and max{m i } = maximum psu size. Here is the procedure: 1. Draw a random number between 1 and N. This indicates which psu you are considering. 2. Draw a random number between 1 and max{m i }; if the random number is less than or equal to M i, then include psu i in the sample; otherwise, go back to step Repeat until the desired sample size is obtained. We know that Lahiri s method results in a pps sample with replacement. (pps stands for probability proportional to size ) Suppose the population has N psu s, with sizes M 1, M 2,, M N. Let X represents the number of pairs of random numbers that must be generated to obtain a sample of size n. Find E[X].