A sociologist is studying the effect of having children within the first 2 years of marriage on the divorce rate.Using hospital birth records,she selects a simple random sample of 200 couples who had children within the first 2 years of marriage.Following up on these couples,she finds that 80 couples are divorced within 5 years.Let p = the population proportion of couples who had children within the first 2 years of marriage and are divorced within 5 years.Is the sample size of n = 200 large enough to estimate p with a margin of error 0.05 with 90% confidence? Use the observed value 0.40 as the value for p^*.

A) Yes
B) No
C) This cannot be determined from the information given.

Question

Christian Ceballos · Accepted Answer

Final Answer : B, Explanation :  To determine if the sample size of n = 200 is large enough to estimate p with a margin of error 0.05 with 90% confidence, we need to calculate the critical value and margin of error as follows:Critical value: z* = 1.645 (from the z-table for 90% confidence)Margin of error: E = z* * sqrt[(p-hat)(1-p-hat)/n]where p-hat = observed value = 80/200 = 0.40E = 1.645 * sqrt[(0.40)(0.60)/200] = 0.065Since the margin of error is 0.065 and we want a margin of error of 0.05 or less, the sample size of n = 200 is NOT large enough to estimate p with a margin of error 0.05 with 90% confidence. Therefore, the answer is B (No).

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