Calculating confidence interval for difference of means | AP Statistics | Khan Academy

Kylie suspected that when people exercise longer, their body temperatures change. She randomly assigned people to exercise for 30 or 60 minutes, then measured their temperatures. The 18 people who exercised for 30 minutes had a mean temperature, so this is the sample mean for that sample of 18 folks, of 38.3 degrees Celsius, with a standard deviation, this is a sample standard deviation for those 18 folks, of 0.27 degrees Celsius.

The 24 people who exercised 60 minutes had a mean temperature of 38.9 degrees Celsius, with a standard deviation. This, once again, these are both sample means and sample standard deviations of 0.29 degrees Celsius. Assume that the conditions for inference have been met and that Kylie will use the conservative degrees of freedom from the smaller sample size.

Which of the following is a 90% confidence interval for the difference in mean body temperature after exercising for the two amounts of time? So pause this video and see if you can figure it out.

All right, now let's work through this together. In previous videos, we talked about the general form of our confidence interval, our t interval, which we're going to use because we're dealing with means and we're dealing with the differences in means. So our t interval is going to have the form: our difference between our sample means, so it could be the sample mean for the 60-minute group minus the sample mean for the 30-minute group, plus or minus our critical t value times our estimate of the sampling distribution of the difference of the sample means.

That is going to be, I think I have enough space here to do it, that is going to be the sample standard deviation of the 60-minute group squared over the sample size of the 60-minute group, plus the sample standard deviation of the 30-minute group squared divided by the sample size of the 30-minute group.

We can actually figure out all of these things. So this is going to be equal to the sample mean for the 60-minute group is 38.9, so it's 38.9 minus the sample mean for the 30-minute group, which is 38.3. 38.3 plus or minus our critical t value.

Now, how do we figure that out? Well, we can use our ninety percent confidence level that we care about, this ninety percent confidence interval. But if we're looking up things on a t-table, we also need to know our degrees of freedom. It says here that Kylie will use the conservative degrees of freedom, and that means that she will look at each of those samples.

So one has a sample size of 18, one has a sample size of 24. Whichever is lower, she will use one less than that as her degrees of freedom. 18 is clearly less lower than 24, so the degrees of freedom in this situation is 18, or r18 minus 1, so 17.

Using that, we can now look this up on a t-table. So our confidence level, 90, and then our degrees of freedom, 17, so that is that row. The 90% confidence level is this column, and so that gives us our critical t value of 1.74.

Going back here, this is going to be plus or minus 1.74 times the square root. What's our sample standard deviation for the 60-minute group? Well, they give it right over here: 0.29, and we're going to have to square that, divided by the sample size for the 60-minute group.

So let's see, the 24 people who exercise for 60 minutes, so divided by 24, plus the sample standard deviation for the 30-minute group. So that's 0.27, 0.27 squared divided by the sample size for the 30-minute group, divided by 18.

And we're done! We can look down at the choices. Let's see, they all got the first part the same because that's maybe the most straightforward part: 38.9 minus 38.3 plus or minus 1.74. So both of these are looking good.

We can rule out these two because they have a different critical t value, and let's see, we have 0.29 squared divided by 24 plus 0.27 squared divided by 18. This one is looking good over here. Let's see, they mixed up our—they put the 30-minute sample size with the sample standard deviation of the 60-minute group, so that won't work.

And so we like choice A.