Based on the results of this hypothesis test, would you expect a confidence interval for the average difference between the reading and writing scores to include 0?

Question

BlackTom AI · Accepted Answer

First, consider what it means to fail to reject H0 in a hypothesis test about the average difference between reading and writing scores. If we fail to reject the null hypothesis that the true mean difference is zero, the data are not strong enough to conclude that the average difference differs from zero.
Option A states: 'Since we failed to reject H0, which claimed the average difference is equal to 0, we would expect a confidence interval to include this value.' This aligns with the fundamental link between hypothesis tests and confidence intervals: failing to reject H0 at a given significance level corresponds to a confidence interval that contains the null value (here, 0) for the parameter.
Option B says we rejected H0 and would expect the CI to include 0. This is inconsistent because rejection of H0 generally implies the CI would exclude the null value, not include it.
Option C posits that we rejected H0, which claimed the average difference is not equal to 0, and yet the CI would include 0. This mixes up the null hypothesis and the alternative; typically, if we reject a two-sided H0 of difference = 0, the CI would not include 0, and the statement about H0 being not equal to 0 is not the standard formulation used in this context.
Option D claims we failed to reject H0 which claimed the average difference is not equal to 0, and the CI would include 0. Here, the null hypothesis is stated as a specific value (0); failing to reject a nonstandard H0 that the difference is not equal to 0 would be an unusual framing, and it does not reflect the conventional interpretation that failing to reject a two-sided H0 of difference = 0 yields a CI that includes 0.
In summary, the correct interpretation is that failing to reject the null hypothesis that the average difference equals zero implies the corresponding confidence interval for the mean difference would include 0, while rejecting H0 would lead to a CI that excludes 0. The other options either misstate the direction of inference or confuse the null with the alternative, leading to incorrect conclusions about the CI containing 0.

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Based on the results of this hypothesis test, would you expect a confidence interval for the average difference between the reading and writing scores to include 0?

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