Question at position 9 A polling organization surveyed 2,002 randomly selected adults who are not scientists and 3,748 randomly selected adults who are scientists. Each adult was asked the question, “Do you think that genetically modified foods are safe to eat?” Of those who are not scientists, 37 percent responded yes, and of those who are scientists, 88 percent responded yes. Which of the following is the standard error used to construct a confidence interval for the difference between the proportions of all adults who are not scientists and all adults who are scientists who would answer yes to the question? 4ST63A polling organization surveyed 2,002 randomly selected adults who are not scientists and 3,748 randomly selected adults who are scientists. Each adult was asked the question, “Do you think that genetically modified foods are safe to eat?” Of those who are not scientists, 37 percent responded yes, and of those who are scientists, 88 percent responded yes. Which of the following is the standard error used to construct a confidence interval for the difference between the proportions of all adults who are not scientists and all adults who are scientists who would answer yes to the question? 4ST63题目解析

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Question at position 9 A polling organization surveyed 2,002 randomly selected adults who are not scientists and 3,748 randomly selected adults who are scientists. Each adult was asked the question, “Do you think that genetically modified foods are safe to eat?” Of those who are not scientists, 37 percent responded yes, and of those who are scientists, 88 percent responded yes. Which of the following is the standard error used to construct a confidence interval for the difference between the proportions of all adults who are not scientists and all adults who are scientists who would answer yes to the question? 4ST63A polling organization surveyed 2,002 randomly selected adults who are not scientists and 3,748 randomly selected adults who are scientists. Each adult was asked the question, “Do you think that genetically modified foods are safe to eat?” Of those who are not scientists, 37 percent responded yes, and of those who are scientists, 88 percent responded yes. Which of the following is the standard error used to construct a confidence interval for the difference between the proportions of all adults who are not scientists and all adults who are scientists who would answer yes to the question? 4ST63题目解析

BlackTom AI · Accepted Answer

We start by identifying what the problem is asking: the standard error to construct a confidence interval for the difference between two proportions, p1 and p2, where p1 is the proportion of non-scientists who answered yes and p2 is the proportion of scientists who answered yes.
Option A (the first candidate expression): sqrt((0.37)(0.63)/2002) + sqrt((0.88)(0.12)/3748). This matches the standard error formula for the difference of two independent proportions: SE = sqrt[p1(1-p1)/n1] + sqrt[p2(1-p2)/n2]. Here p1 = 0.37, n1 = 2002, p2 = 0.88, n2 = 3748. The plus sign is appropriate because the variance of the difference p̂1 - p̂2 is Var(p̂1) + Var(p̂2) when the samples are independent, so we add the two standard errors in the SE expression.
Option B (the second candidate): sqrt((0.37)(0.63)/2002) − sqrt((0.88)(0.12)/3748). This uses a minus sign between the two standard error components. That would imply subtracting the uncertainty of one group from the other, which is not correct for the SE of a difference in proportions, since the variances add for independent samples. Therefore this is not the correct SE form.
Option C (the third candidate): sqrt((0.70)(0.30)/2002) + sqrt((0.70)(0.30)/3748). This uses a pooled proportion of 0.70 (which would arise if we pooled the two samples under a null hypothesis of equal proportions). However, the standard error for a difference of two proportions without pooling uses the distinct p1 and p2 in each term, not a single pooled p. Therefore this is not the correct SE for the difference between the two proportions given.
Option D (the fourth candidate): (0.37)(0.63)/2002 + (0.88)(0.12)/3748, all under a square root with no division or further structure shown. This resembles a misformatted or incomplete version of the SE formula, lacking the required square roots of each term (i.e., it mixes the components incorrectly inside a single square root). The correct SE uses two separate square-root terms added together, not a single square root of a sum of fractions.
Option E (the fifth candidate): sqrt((0.37)(0.63)/2002) + sqrt((0.88)(0.12)/3748) — identical in substance to Option A but if presented differently could be a duplicate or misformatted in some displays. The key idea is that the correct structure is two separate square roots added together, each with its own p(1−p)/n term, as in Option A.
To summarize, the essential concept is that the standard error for the difference between two independent proportions is obtained by adding the standard errors from each group: SE = sqrt[p1(1−p1)/n1] + sqrt[p2(1−p2)/n2], with p1 = 0.37, n1 = 2002, p2 = 0.88, n2 = 3748. Any expression deviating from this structure—especially using a minus sign, pooling, or a single pooled square root—fails to reflect the correct variance composition for the difference of two independent proportions.

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