(14.29) The birth control shot is one of the most effective methods of birth con
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Question
(14.29) The birth control shot is one of the most effective methods of birth control available, and it works best when you get the shot regularly, every 12 weeks. Under ideal conditions, only 1% of women getting the shot become pregnant within one year. In typical use, however, 3% become pregnant. A study of the effectiveness of the birth control shot interviews a random sample of 600 women who are using the shot as their method of birth control.
Step 1: Based on the information about typical use, we are interesting in finding the probability that at least 20 of these women become pregnant in the next year. Is the Normal approximation permissible? No, because p is very far from 0.5. No, because np and n(1 p) differ from each other too much. Yes, because np and n(1 p) are not smaller than 10. Yes, because the sample is large. Yes, because we may reasonably assume that the population is Normal.
Step 2: What is the probability that at least 20 of these women become pregnant in the next year? 0.25 0.6915 0.3156 0.2389
Step 3: We can't use the Normal approximation to the binomial distributions to find this probability under ideal conditions. Why not? Because np and n(1 p) differ from each other too much. Because the Normal approximation gives an unreasonable small probability. Because p is close to zero. Because np is less than 10.
Explanation / Answer
step 1: Typical condition. n = 600, p = 0.03
np = 600 * 0.03 = 18, n(1-p) = 600 * 0.97 = 582
The Normal approximation can be used. Yes, beacuse np and n(1 p) are not smaller than 10
step2: = np 600*.03 = 18, The standard deviation is s = sqrt(n*p*(1 - p)) = sqrt(600*.03*.97) = 4.1785
t statistic = (20 - 25)/(4.1785/sqrt(600)) = -29.3105, df = 599
Probability that at least 20 of these women become pregnant = P(X20) = P(z (20-18)/4.1785) = P(z 0.47864)
= 1 - P(z < 0.47864) = 1 - 0.68439 (as the area in table is to the left of z score)= 0.3156
step 3: p = 0.01, np = 600 * 0.01 = 6
We can’t use the Normal approximation for the ideal case because np = 6.
Because np is less than 10.
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