Conclusion for a two-sample t test using a confidence interval | AP Statistics | Khan Academy

Yuna grows two varieties of pears: bosk and anju. She took a sample of each variety to test if their average caloric contents were significantly different. Here is a summary of her results, or here is a summary of her results, and so they give the same data for both of these samples. Once again, they happen to have the same sample size. They don't need to, but over here, instead of giving us a p-value, we've gotten a confidence interval.

Yuna wants to use these results to test her null hypothesis that the mean caloric content is the same versus her alternative hypothesis that they are different. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding p-value and conclusion at the alpha equal to 0.01 level of significance?

So pause this video and see if you can figure that out. Remember what a 99% confidence interval is: that says that if we construct confidence intervals 100 times, then 99 of those times we should overlap with the true parameter that we're trying to estimate. In that case, the true parameter is the true difference of these means.

Now, when we do a hypothesis test, we always start assuming that the null hypothesis is true. If we assume that the null hypothesis is true, another way of writing this null hypothesis is that the two means are equal. That's the same thing as the difference of the means equaling zero. Since we're assuming this and this is a 99% confidence interval, then 99 out of 100 times we do this, we should see that this interval overlaps with what we're assuming is the true parameter.

Right over here, this interval does indeed overlap with zero. If you take 4 minus 6.44, you're going to get negative 2.44. So 0 is definitely in the interval. Another way to think about it is we're not in the one percent of the times where we don't overlap. If we were in the one percent of times where we don't overlap with the assumed difference, then we would reject the null hypothesis.

Another way to think about it is our significance level, 0.01. Right over here, it's one minus our confidence level. If our 99% confidence interval overlaps with μ from the bosk pairs minus the mean caloric content of the anju pairs equaling 0, then that means that the p-value is greater than 0.01. We could also say that our p-value is greater than our significance level because that is our significance level.

Because of that, we fail to reject our null hypothesis. If this did not overlap with our assumed difference in the means, if it did not overlap with zero, then we would be in that one in a hundred scenario. That would tell us that our p-value is less than 0.01; our p-value is less than one minus our confidence level. In that case, we would reject the null hypothesis, and it would suggest that there is a difference in caloric content.

But because we fail to reject it, we can't conclude that there's a difference in caloric contents.