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Quiz 2 Solutions

# Quiz 2 Solutions - d What is the name and parameters of the...

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STA3032 QUIZ 2 Solutions SPRING 2012 Problem 5.28 (modified): The US statistical Abstract reports that, the median family income, in the US, in 1989, was \$34,200. Suppose 4 families were randomly selected from the population of all families in the US in 1989. Let X = number of families in the sample that have incomes above the median of \$34,200. Answer the following (Parts c to e are added): a) Find the probability that, among the 4 families selected, all had incomes above the median, that is find P(X = 4) b) Find the probability that, among the 4 families selected, one had income below the median . If one of the 4 families had income below the median, then the remaining 3 families had incomes above the median, i.e., X = 3. Hence, we are asked to find P(X = 3). c) Write the formula you have used in parts (a) and (b), for p(x), the pmf of X.
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Unformatted text preview: d) What is the name and parameters of the distribution of X? X has the binomial distribution with parameters n = 4 and π = ½, i.e., X ~ B(4, ½ ). e) Give reasons for your answer to part (e). i. There is a Bernoulli experiment with “Success” defined as selecting a family that has income above the median. ii. The experiment is repeated n = 4 times (selection of 4 families, assumed to be one at a time) iii. The repetitions are independent (random sampling guarantees that). iv. Probability of “Success” is ½ since the median divides the population units into two equal parts. v. The random variable of interest counts the number of “Success”s, i.e., families that have income above the median. Hence X has the binomial distribution with parameters n = 4 and π = ½, i.e., X ~ B(4, ½ )...
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