Confidence Intervals for Two Populations

Summary:  Confidence intervals for two populations are easy enough to calculate on your TI-83. But one or both endpoints can be negative, and that means you need to interpret your numbers carefully. You must also distinguish between mean difference (for paired data) and difference in means (for unpaired data).

Dependent Means / Paired Data

Just as you do a simple Student’s t test of a hypothesis about paired data, you compute a simple TInterval to estimate the mean difference.

Example A: Coffee and Heart Rate

Dabes & Janik's Statistics Manual (1999) page 264 shows the change in heart rate for six subjects after drinking coffee:

To compute a 95% CI, proceed just as you would for any other TInterval: enter the numbers in L1, then press [STAT] [◄] [8]. The input and output screens are shown at right.

Conclusion: With 95% confidence, the mean increase in heart rate for all people after drinking coffee is between 2.2 and 6.1 beats per minute. (Notice that this is the mean difference μ_A−B, not a difference in means μ_A−μ_B. With paired data you are predicting the mean difference between two measurements taken from one randomly selected individual.)

Example B: Coffee and Heart Rate with Negatives

Now let’s alter the data a bit to bring up a new concept:

Notice that the third person’s heart rate declined after drinking coffee. Now when you compute a 95% CI you get the results shown at right.

How should you interpret a negative endpoint in the interval? Remember that you are computing a CI for the quantity After−Before. You could follow the earlier pattern and say “With 95% confidence, the mean increase in heart rate for all individuals after drinking coffee is between −1.2 and 5.8 beats per minute,” but only a mathematician would love a statement that talks about an increase being negative. Instead, you draw attention to the fact that the change might be a decrease or an increase, as follows.

Conclusion: With 95% confidence, the mean change in heart rate for all individuals after drinking coffee is between a decrease of 1.2 and an increase of 5.8 beats per minute. Since it’s obviously very important to get the direction right, be sure to check your conclusion against your H₁ and your original definition of d.

Remark 1: Though it’s correct to present the CI as a point estimate and margin of error, it’s probably not a good idea because that form is so easy to misinterpret. If you say “With 95% confidence, the mean change in heart rate for all individuals is 2.3±3.5 beats per minute,” many people won’t notice that the margin of error is bigger than the point estimate, and they’ll come to the false conclusion that you have established an increase in heart rate after drinking coffee. As statistics mavens we have a responsibility to present our results clearly, so that people draw the right conclusions and not the wrong ones.

Remark 2: Remember that the CI occupies the middle of the distribution while the HT looks at the tails. If 0 is inside the CI, it can’t be in either tail. Therefore, from this confidence interval you know that testing the null hypothesis μ_d = 0 at the 0.05 level (0.05 = 1−95%) would fail to reject H₀: the experiment failed to find a significant difference in heart rate after drinking coffee. A CI will always match a two-tailed HT in this way.

Remark 3: Remember the difference between “no significant difference found” and “no difference exists”. Since 0 is in the CI, you can’t say whether there is a difference. The correct statement, “I don’t know whether there is a difference,” is different from the incorrect “There is no difference.”

Independent Means / Unpaired Data

With unpaired data, you have a sample from one population and an independent sample from a different population. Your HT would be about a difference in means between the two populations, and so is your CI.

Example C: Heights of Men and Women

Page 425 of Johnson & Kuby’s Just the Essentials of Elementary Statistics 3/e presents an example of heights of randomly selected men and women at a college, and asks you to estimate the difference μ_M − μ_F in a 95% CI.

The TI-83 computes μ₁ − μ₂, and therefore you must define males as population 1 and females as population 2. On your TI-83, press [STAT] [◄] and scroll down to find 0:2-SampTInt. Enter the sample statistics and use Pooled:No. Input and output screens are shown at right.

Conclusion: With 95% confidence, the average man at that college is between 4.8″ and 7.2″ taller than the average woman, or μ_M−μ_F = 6.0±1.2”.

Remark 1: The difference from the case of dependent means is subtle but important. With dependent means (paired data), the CI was about the average difference in measurements of a single randomly selected individual. But with independent means (unpaired data), the CI is about the difference between the averages for two different populations.

Remark 2: You could perform a 2-sample F test and use it to decide whether to pool data in the 2-sample t interval, but it rarely makes a major difference in the results.

Example D: GPA of Fraternity Members and Nonmembers

Johnson & Kuby’s Just the Essentials of Elementary Statistics 3/e presents another example on page 427. What is the difference (if any) in academic performance between fraternity members and nonmembers? Forty members of each population were randomly selected, and their cumulative GPA recorded as an indication of performance. The results were as follows:

The CI is −0.46 to +0.10, with 95% confidence. Remember that the TI-83 computes a CI for μ₁ − μ₂. Since fraternity members are population 1, the CI says that the true difference in academic performance is somewhere between 0.46 worse and 0.10 better for fraternity members relative to nonmembers, with 95% confidence. You can also express that as somewhere between 0.46 in favor of nonmembers and 0.10 in favor of fraternity members, with 95% confidence.

Remark 1: Don’t be fooled by the fact that the CI is mostly below zero. You really cannot conclude that fraternity members probably have lower academic performance. Remember that the 95% CI is the result of a process that captures the true population mean (or difference, in this case) 95 times out of 100. But you can’t know where in that interval the true mean (or difference) lies. If you could, there would be no point to having a CI!

Remark 2: Even though zero is within the CI, you must not say that there is no difference in performance between members and nonmembers. The difference might indeed be zero, but it might also be anywhere between 0.46 in one direction and 0.10 in the other. There’s even a 5% chance that the true difference lies outside those limits. Always bear in mind the difference between insufficient evidence for and evidence against.

Proportions

Example E: Opinion Poll

The following data are from Dabes & Janik's Statistics Manual (1999) page 269. Men and women were polled on whether they favored legalized abortion, and the results were as follows:

What is the 98% CI for the difference in support between females and males? The TI-83 will compute an interval for p₁−p₂; designate females as population 1.

On the TI-83, press [STAT] [◄] and scroll up to find B:2-PropZInt. The input and output screens are shown below.

Conclusion: To begin with, we are 98% confident that the true difference in support, females minus males, is between −7.3% and +27.3%. But we can do better than that. We can see that the true difference is somewhere between females are 7.3% less in favor than males and females are 27.3% more in favor, with 98% confidence. So our final statement is

We are 98% confident that somewhere between 7.3% fewer females than males, and 27.3% more females than males, support legalized abortion.

You might be interested to compare the formula for the margin of error for a proportion in one population with the formula for the margin of error for the difference of proportions in two populations