The gap between men and women with college degrees has been narrowing in recent years, with women now more likely to hold a bachelor's degree than men. To investigate this, a researcher sampled 128 males and 122 females aged 25 and older and asked if they have a four-year college degree. The data showed that while 49 females had a four-year college degree, only 46 males did. The researcher wants to determine the difference between the two groups' proportions of having a four-year college degree is significant. Which proportion(s) must be used when calculating the standard error for this inference?

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BlackTom AI · Accepted Answer

We start by identifying the context: comparing two independent proportions (males vs. females) and determining what to use in the standard error for the difference in proportions.
Option 1: '0.40 and 0.36' — These numbers are the rounded sample proportions for each group ( females ≈ 49/122 ≈ 0.4016, males ≈ 46/128 ≈ 0.3594 ). While these are the observed group proportions, the standard error for a two-proportion test under the null hypothesis of equal proportions relies on the pooled proportion, not on the separate sample proportions.
Option 2: '0.38' — This is the pooled proportion calculated from both groups together: p_hat = (x1 + x2) / (n1 + n2) = (49 + 46) / (122 + 128) = 95 / 250 = 0.38. In the standard error formula for the difference in proportions, we use p_hat to estimate the common probability under H0, and then compute SE as sqrt(p_hat*(1 - p_hat)*(1/n1 + 1/n2)). This makes sense because under the null hypothesis that the two population proportions are equal, a single pooled proportion should describe the success probability across both samples.
Option 3: '0.195' — This value resembles half of 0.39, but it has no direct interpretive role in the standard error formula for two proportions. It does not correspond to the pooled estimate nor to the individual group proportions and seems unrelated to the SE calculation here.
Option 4: '0.383 and 0.377' — These look like rounded individual sample proportions for the two groups (female ≈ 0.4016 rounds to 0.38–0.40, male ≈ 0.3594 rounds to 0.36–0.37). However, the standard error for the difference in proportions does not use both rounded group proportions separately; it specifically uses the pooled proportion to reflect the null hypothesis of equal proportions. Using the separate group proportions would misrepresent the SE under H0.
In summary, the standard error for the inference about the difference in two proportions under the usual test framework relies on a single pooled estimate of the proportion, not the individual group proportions. Therefore, the only option that matches the standard approach is the pooled proportion value.

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