In order to gain additional information about respondents, some marketing resear

Question

In order to gain additional information about respondents, some marketing researchers have used ultraviolet ink to precode questionnaires that promise confidentiality to respondents. Of 173 randomly selected marketing researchers who participated in an actual survey, 138 said that they disapprove of this practice. Suppose that, before the survey was taken, a marketing manager claimed that at least 82 percent of all marketing researchers would disapprove of the practice.

(a) Assuming that the manager's claim is correct, calculate the probability that 138 or fewer of 173 randomly selected marketing researchers would disapprove of the practice. Use the normal approximation to the binomial. (Round z value to 2 decimal places. Round your answer to 5 decimal places.)

P(x 138)

Explanation / Answer

Solution

Normal approximation to Binomial

If X B(n, p), (X – np)/{np(1 - p)} N(0, 1), provided np 5 and np(1 - p) 5 and

P(X < t) is approximated by P[Z < (t – np)/ {np(1 - p)}]

In the given problem, n = 173, p = 0.82 (i.e., 82%).

Since both np = 141.86 and np(1 - p) = 25.5348 are greater than 5, Normal approximation is permitted.

P(X < 138) = P[Z < {(138 – 141.86)/25.5348}] = P(Z < - 0.76) = 0.22263 ANSWER

If X B(n, p), (X – np)/{np(1 - p)} N(0, 1), provided np 5 and np(1 - p) 5 and

P(X < t) is approximated by P[Z < (t – np)/ {np(1 - p)}]

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