A two-sample \(t\)-test of the hypotheses \(H_0 : \mu_1 - \mu_2 = 0\) versus \(H_a : \mu_1 - \mu_2 \gt 0 \) produces a \(p\)-value of 0.03. Which of the following must be true?

1. A 90 percent confidence interval for the difference in means will contain the value 0.
2. A 95 percent confidence interval for the difference in means will contain the value 0.
3. A 99 percent confidence interval for the difference in means will contain the value 0.

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Since this is a one-tailed test, the critical region (area under the curve) would be present in the right of the distribution with an area of 0.03.

Confidence intervals are two tailed. If we have a 90 percent confidence interval, there are will be 0.05 region in the left and right tails of the distribution. Therefore, the 90 percent confidence interval does not necessarily contain the value 0.

As the confidence interval increases, the critical region will begin to overlap, implying that the \(p\)-value is significant.

At 95% confidence, the tail regions each comprise 0.025 area. The \(p\)-value of 0.03 is within the confidence interval rather than outside of it. A similar thing occurs in the 99 percent confidence interval.