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## Testing for the difference of two population means

# Conclusion for a two-sample t test using a confidence interval

AP.STATS:

UNC‑4 (EU)

, UNC‑4.AA (LO)

, UNC‑4.AA.1 (LO)

## Video transcript

- [Voiceover] Yuna grows
two varieties of pears, Bosc and Anjou. She took a sample of each variety to test if their average caloric contents were significantly different. Here is a summary of her results. And so they give the same data
for both of these samples, and once again they happen
to have the 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 is 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, that 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. And so if we assume that
the null hypothesis is true, well another way of writing
this null hypothesis, if the two means are equal, that's the same thing as the difference of the means equaling zero. And since we're assuming this, and this a 99% confidence interval, then 99 out of 100 times that we do this, we should see that this interval overlaps with what we're assuming
is the true parameter right over here. Now this interval does
indeed overlap with zero, if you take four minus 6.44, you're gonna get negative 2.44. So zero is definitely in the interval. And so another way to think about it, we're not in the 1% of the times where we don't overlap. If we were in the 1% of
times where we don't overlap with the assumed difference, then we would reject the null hypothesis. Or another way to think about it is our significance level
0.01 right over here, it's one minus our confidence
level, right over here. If our 99% confidence interval overlaps with mu from the Bosc pears minus the mean caloric content of the Anjou pears, equaling zero, then that means, that means that the P-value is greater than 0.01. And so we could also
say that our p-value is greater than our significance level, because that is our significance level, and 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 and then that would tell
us that hey our p-value is less than 0.01. Our P-value is less than one
minus our confidence level and in that case we would
reject the null hypothesis and it would suggest that there is a difference in caloric content. But because we failed to reject it, we can't conclude that there's a difference in caloric content.

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