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# Calculating confidence interval for difference of means

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.V (LO)
,
UNC‑4.V.1 (EK)
,
UNC‑4.V.2 (EK)
,
UNC‑4.X (LO)
,
UNC‑4.X.1 (EK)
,
UNC‑4.X.2 (EK)
,
UNC‑4.Y (LO)
,
UNC‑4.Y.1 (EK)
,
UNC‑4.Y.2 (EK)

## Video transcript

keilly 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 point three degrees Celsius with a standard deviation this is a sample standard deviation for those 18 folks of zero point two seven degrees Celsius the 24 people who exercised sixty minutes had a mean temperature of 38 point nine degrees Celsius with a standard deviation this once again these are both sample means and sample standard deviations of zero point two nine 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 alright now let's work through this together so 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 and 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 and that is going to be I think 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 and so 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 thirty eight point nine so it's thirty eight point nine 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 90 percent confidence level that we care about this 90% confidence interval but if we're looking up things on a tea table we also need to know our degrees of freedom and 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 which ever 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 our 18 minus 1 so 17 and so using that and that we can now look this up on a tea 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 point 7 4 so going back here this is going to be plus or minus 1 point 7 4 times the square root times the square root what's our sample standard deviation for the 60 minute group well they give it right over here zero point two nine and we're gonna have to square that divided by the sample size for the 60 minute group so let's see the 24 people who exercised for 60 minutes so divided by 24 plus the sample standard deviation for the 30 minute group so that's zero point two seven zero point two seven squared divided by the sample size for the 30 minute group divided by 18 and we're done and 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 thirty-eight point thirty eight point nine minus thirty eight point three plus or minus one point seven four so both of these are looking good we can rule out these two because they have a different critical T value now let's see we have 0.29 squared divided by 24 plus 0.2 7 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
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