1. Chapter 25
Asking and Answering Questions About the Difference Between Two Population Means: Paired Samples

2. Estimating the Difference Between Two Population Means: Paired-Samples t Confidence Interval for a Difference in Population Means

3. The Paired-Samples t Confidence Interval for a Difference in Population Means
Appropriate when the following conditions are met:
The samples are paired (for example, before/after data.
The n sample differences can be viewed as a random sample from a population of differences (or it is reasonable to regard the sample of differences as representative of the population of differences).
The number of sample differences is large (n ≥ 30) or the population distribution of differences is approximately normal.

4. The Paired-Samples t Confidence Interval for a Difference in Population Means
When these conditions are met, a confidence interval for the difference in population means is
𝑥 𝑑 ± 𝑡 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠 𝑑 𝑛
Where
n = the number of sample differences,
𝑥 𝑑 = mean of the n sample differences,
𝑠 𝑑 = standard deviation of the n sample differences.
The t critical value is based on df = n – 1.

5. The Paired-Samples t Confidence Interval for a Difference in Population Means
𝑥 𝑑 ± 𝑡 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠 𝑑 𝑛
Interpretation of Confidence Level The confidence level specifies the long-run proportion of the time that this method is successful in capturing the true, but unknown, difference d = 1 - 2 in population means. Interpretation of Confidence Interval You can be confident that the computed interval contains the true, but unknown, value of the difference d = 1 - 2 in the population. This statement should be worded in context.

6. Benefits of Ultrasound
In a study to investigate the effect of ultrasound therapy on knee extension, range of motion was measured for people in a representative sample of physical therapy patients both before and after ultrasound therapy.
Because the samples are paired, the first thing to do is to compute the sample differences.
Range of Motion
Patient 1 2 3 4 5 6 7
Before Ultrasound 31 53 45 57 50 43 32
After Ultrasound 59 46 64 49 40
Differences -1 -6 -8 -2 1 -7
The mean and standard deviation computed from these sample differences are 𝑥 𝑑 = and sd = 3.51

7. Range of Motion Continued . . .
Patient 1 2 3 4 5 6 7
Before Ultrasound 31 53 45 57 50 43 32
After Ultrasound 59 46 64 49 40
Differences -1 -6 -8 -2 1 -7
The mean and standard deviation computed from these sample differences are 𝑥 𝑑 = and sd = 3.51
You want to estimate
md = m1 – m2 = mean difference in knee random of motion
where
m1 = mean knee range of motion before ultrasound
and
m2 = mean knee range of motion after ultrasound

8. Range of Motion Continued . . . Calculate
The t critical value for df = 6 and 95% confidence level is 2.45.
𝑥 𝑑 ± 𝑡 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠 𝑑 𝑛 =−3.43±
=(−6.68, −0.18)
Communicate Results
We are 95% confident that the interval (-6.68, -0.18) contains the true, but unknown difference in mean range of motion. Because both endpoints are negative, you would estimate that knee range of motion after ultrasound is greater than the mean range of motion before ultrasound by somewhere between 0.18 and 6.68 degrees.
The method used to construct this interval is successful in capturing the actual difference in population means about 95% of the time.

9. The Paired t Test Appropriate when the following conditions are met:
The samples are paired.
The n sample differences can be viewed as a random sample from a population of differences (or it is reasonable to regard the sample of differences as representative of the population of differences).
The number of sample differences is large (n ≥ 30) or the population distribution of differences is approximately normal.

10. Summary of the Paired t test for Comparing Two Population Means Continued
When these conditions are met, the following test statistic can be used:
𝑡= 𝑥 𝑑 − 𝜇 0 𝑠 𝑑 𝑛
Where m0 is the hypothesized value of the population mean difference from the null hypothesis, n is the number of sample differences, and 𝑥 𝑑 and 𝑠 𝑑 are the mean and standard deviation of the sample differences.

11. Summary of the Paired t test for Comparing Two Population Means Continued
Form of the null hypothesis: H0: md = m0
When the conditions are met and the null hypothesis is true, this t test statistic has a t distribution with df = n – 1.
When the Alternative Hypothesis Is . . .
The P-value Is . . .
Ha: md > m0
Area under the t curve to the right of the calculated value of the test statistic
Area under the t curve to the left of the calculated value of the test statistic
Ha: md < m0
2·(area to the right of t) if t is positive Or
2·(area to the left of t) if t is negative
Ha: md ≠ m0

12. In a study to investigate the effect of ultrasound therapy on knee extension, range of motion was measured for people in a representative sample of physical therapy patients both before and after ultrasound therapy.
Because the samples are paired, the first thing to do is to compute the sample differences.
Range of Motion
Patient 1 2 3 4 5 6 7
Before Ultrasound 31 53 45 57 50 43 32
After Ultrasound 59 46 64 49 40
Differences -1 -6 -8 -2 1 -7
Is there evidence that the ultrasound therapy increases range of motion?
The mean and standard deviation computed from these sample differences are
𝑥 𝑑 = and sd = 3.51

13. md = m1 – m2 = mean difference in range of motion (before – after)
Range of Motion Continued . . .
Hypotheses:
The population characteristics of interest are
m1 = mean range of motion for physical therapy patients before ultrasound
m2 = mean range of motion for physical therapy patients after ultrasound
Because the samples are paired, you should also define md :
md = m1 – m2 = mean difference in range of motion (before – after)
Translating the question of interest into hypotheses gives:
H0: md = 0 versus H0: md < 0

14. Range of Motion Continued . . . Calculate
Test Statistic: 𝑡= −3.43− =−2.59
P-value:
P-value = area under t distribution with 6df to the left of -2.59
= 0.02
Communicate Results
Because the P-value (0.02) is less than 0.05, reject H0. There is convincing evidence that mean knee range of motion for physical therapy patients before ultrasound is less than the mean range of motion after ultrasound.

15. Avoid These Common Mistakes

16. Avoid These Common Mistakes
Remember that the results of a hypothesis test can never show strong support for the null hypothesis. In two-sample situations, this means that you shouldn’t be convinced that there is not difference between two population means based on the outcome of a hypothesis test.

17. Avoid These Common Mistakes
If you have complete information (a census) for both populations, there is no need to carry out a hypothesis test or to construct a confidence interval – in fact, it would be inappropriate to do so.

18. Avoid These Common Mistakes
Don’t confuse statistical significance with practical significance. In the two-sample setting, it is possible to be convinced that two population means are not equal even in situations where the actual difference between them is small enough that it is of no practical use.
After rejecting a null hypothesis of no difference, it is useful to look at a confidence interval estimate of the difference to get a sense of practical significance.

19. Avoid These Common Mistakes
Correctly interpreting confidence intervals in the two-sample case is more difficult than in the one-sample case, so take particular care when providing two-sample confidence interval interpretations.
Because the two-sample confidence interval estimates a difference (m1 – m2), the most important thing to note is whether or not the interval includes 0.