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Unformatted text preview: 0.1 Exercises for Chapter 6 1. The financial manager of a large department store chain selected a random sample of 200 of its credit card customers and found that 136 had incurred an interest charge during the previous year because of an unpaid balance. (a) What do you think is the parameter of interest in this study? (b) Calculate a point estimate for the parameter of interest listed in part (a). 2. Suppose a person is selected at random and is given a hand-writing-recognition-equipped Apple Newton computer to try out. The Apple Newton is fresh from the manufacturer, and the person is asked to write “I think Apple Newton’s are neat”. But often the Apple Newton mis-interprets the words and types back something quite different (we needn’t go into all the possibilities here). A computer magazine conducts a study to estimate the probability, p , that, for a randomly selected owner of a hand-writing-recognition-equipped Apple Newton computer, the phrase “I think Apple Newton’s are neat” will be mis-interpreted. Suppose that they will observe X mistakes in n tries ( n owners). (a) What is your estimator for p ? What properties does this estimator have? Are there any general principles that support the use of this estimator? (b) What is your estimate ˆ p if the magazine finds 33 mistakes in 50 tries? (c) Now suppose that the magazine’s pollster recommends that they also estimate the stan- dard error of the estimator ˆ p . For the given data, what is your estimate of the standard error, and how do you justify this choice? 3. A food processing company is considering the marketing of a new product. Among 40 ran- domly chosen consumers 9 said that they would purchase the new product and give it a try. Estimate the true proportion of potential buyers, and state its standard error. 4. Let X 1 ,...,X n be a sample from U (0 ,θ ). Find the moments estimator of θ . 5. For a sample of 6 jugs of 2 % lowfat milk produced by “Happy Cow Dairy” the fat content X i has been determined as: 2 . 08 2 . 10 1 . 81 1 . 98 1 . 91 2 . 06 (a) Making no assumptions on the distribution of the fat content, estimate the proportion of milk jugs having a fat content of 2.05 % or more. (b) Making no assumptions on the distribution of the fat content, estimate the mean fat content of “Happy Cow Dairy” 2 % lowfat milk. (c) Making no assumptions on the distribution of the fat content, (i) your estimator in part 5a is (circle all correct statements): Unbiased Maximum likelihood estimator Moments estimator (ii) your estimator in part 5b is (circle all correct statements): Unbiased Maximum likelihood estimator Moments estimator (d) Assuming normality, the maximum likelihood estimators of the mean and variance are given by X and [( n- 1) /n ] s 2 , respectively. Calculate the maximum likelihood estimator of the proportion of milk jugs having a fat content of 2.05 % or more. [Hints: a) P ( X > x ) = 1- Φ(( x-...
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Penn State.
- Fall '00