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Unformatted text preview: Stat 651 (Winter 2011) Kaizar Homework 4 Solutions 40 points total Exercises 1. (9 points) (a) I can take a SRS of the chickens. I find the total weight (with the bones and fat and skin) of each of the sampled chickens, which I denote x i . I also find the useable weight (without the bones and fat and skin) of each of the sampled chickens, which I denote y i . Finally, I find the total weight of all the chickens t x = ∑ N i =1 x i . I can now use this information to find the ratio estimate: ˆ t y,r = ˆ N ¯ y r = hatwidest parenleftBigg ∑ N i =1 x i 1 N ∑ N i =1 x i parenrightBigg ¯ y = hatwidest parenleftbigg t x ¯ x U parenrightbigg ¯ y = parenleftbigg t x ¯ x parenrightbigg ¯ y = t x ¯ y ¯ x = t x ˆ B In this case, ratio estimation is useful because we do not know the size of the population ( N ), and so we can use ratio estimation to estimate the total. Without ratio estimation, we do not have any methods to find the total. (b) Because actual age is difficult to measure, I would only take the core and measure the age of the tree ( y i ) for a SRS sample of trees. However, I would measure diameter ( x i ) on all the trees. Then, I could use a ratio estimator of the population mean: ˆ ¯ y r = ¯ y t x ¯ x Using a ratio estimator would be beneficial because I believe that the correlation between the diameter and age of a tree is positively correlated and has a relatively straight line relationship through the origin (when the tree is zero years old, the diameter is also zero). Thus, I believe the variance of this estimator is smaller than if I had simply used ¯ y (c) In this case, I would want to find the total time devoted to sports divided by the total time devoted to the news shows (without the commercials). Since ABC and CBS broadcast news 3 times a day, NBC broadcasts news twice a day, and Fox and PBS both broadcast news once a day, it would be prohibitively expensive to do a census of these news programs 1 over the 365 days of the year. Instead, I can take a SRS of the news shows and record the total length of the broadcast ( x i ) and the length of the sports segment for the broadcast ( y i ) for each of the chosen shows. I would then estimate the population proportion using a ratio estimator: ˆ B = ˆ t y ˆ t x Using a ratio estimator is really the only option for this question, since the estimand is a population ratio. 2. (6 points) (a) For this diamonds example, the sample correlation between the carat weight and the price is quite large and positive at 0.94. From the plot below there is perhaps some little bit of curve in the line, and the intercept is a little lower than zero. Because the intercept may not equal zero, a ratio estimate may be biased and so a regression estimator may be better....
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This note was uploaded on 07/26/2011 for the course STA 651 taught by Professor Kaizar during the Winter '11 term at Ohio State.
- Winter '11
- Sampling Methods