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# Hypergeometric Distribution and Confidence Intervals

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Suppose that in a population of 10 items, 3 are defective and 7 are not. Suppose that two items are chosen at random for inspection. Let X be the number of defective items inspected. report all probabilities to a minimum of 5 decimal places of accuracy.

(a) Clearly explain why X does not have a binomial distribution. [Hint: Which of the binomial conditions is not satisfied here?]
(b) What kind of distribution does X have? List all possible values of X and their probabilities.
(c) Now suppose that instead of sampling from a population of 10 items with 3 defective, we sample from a population of 100 items with 30 defective and 70 not. suppose that two items are chosen at random for inspection. Let X2 be the number of defective items inspected. What kind of distribution does X2 have? Find the complete probability distribution of X2.
(d) repeat(c) for a population of size 1000, with 300 defective and 700 not. (e) repeat(c) for a population of size 10,000, with 3000 defective and 7000 not. (f) Determine the probability distribution for binomial distribution with n = 2 and p = 0.3. how does this distribution compare to the ones found in parts (b),(c),(d) and (e)? explain in much detail as possible why this makes sense.