Comparing the Means of Two Dependent Populations The Paired T-test ….

Assumptions: 2-Sample T-Test Data in each group follow a normal distribution. For pooled test, the variances for each group are equal. The samples are independent. That is, who is in the second sample doesn’t depend on who is in the first sample (and vice versa).

What happens if samples aren’t independent? That is, they are “dependent” or “correlated”?

Do males earn higher average starting salaries than females? (in $1,000s) Males Females 60 32 32 44 80 22 50 40 Sample Average: $55.5 $34.5 Real question is whether males and females in the same job earn different average salaries. Better then to compare the difference in salaries in “pairs” of males and females.

Now, a Paired Study Salaries (in $1,000s) Job Males Females Difference=M-F Non-Profit 22 20 2.0 Education 29 28 1.0 Doctor 80 78 2.0 Scientist 35 32 3.0 Averages 41.5 39.5 2.0 P-value = How likely is it that a paired sample would have a difference as large as $2,000 if the true difference were 0? Problem reduces to a One-Sample T-test on differences!!!!

Hypotheses for Paired T-test Does the average difference of the population, D, differ from 0? Null hypothesis: H0: D = 1 - 2 = 0 Alternative hypotheses: HA: D = 1 - 2  0 HA: D = 1 - 2 > 0 HA: D = 1 - 2 < 0

The Paired-T Test Statistic If: there are n pairs and the differences are normally distributed Then: The test statistic, which follows a t-distribution with n-1 degrees of freedom, gives us our p-value:

The Paired-T Confidence Interval If: there are n pairs and the differences are normally distributed Then: The confidence interval, with t following t-distribution with n-1 d.f. estimates the actual population difference:

Data analyzed as 2-Sample T Two sample T for M vs F N Mean StDev SE Mean M 4 41.5 26.2 13 F 4 39.5 26.1 13 95% CI for mu M - mu F: ( -43, 47) T-Test mu M = mu F (vs not =): T = 0.11 (=2/18.5) P = 0.92 DF = 6 Pooled StDev = 18.5 P = 0.92. Do not reject null. Insufficient evidence to conclude that average starting salaries differ between males and females.

Data analyzed as Paired T Paired T for M - F N Mean StDev SE Mean M 4 41.5 26.2 13.1 F 4 39.5 26.1 13.1 Difference 4 2.000 0.816 0.408 95% CI for mean difference: (0.701, 3.299) T-Test of mean difference = 0 (vs not = 0): T-Value = 4.90 (=2/0.408) P-Value = 0.016 P = 0.016. Reject null. Sufficient evidence to conclude that average starting salaries differ between males and females.

What happened? P-value from two-sample t-test is just plain wrong. (Assumptions not met.) We removed or “blocked out” the extra variability in the data due to differences in jobs, thereby focusing directly on the differences in salaries. The paired t-test is more “powerful” because the paired design reduces the variability in the data.

What is the effect of alcohol on useful consciousness? Ten male subjects taken to a simulated altitude of 25,000 ft and given tasks to perform. For each, time (in seconds) at which “useful consciousness” ended was recorded. 3 days later, experiment was repeated one hour after subjects ingested 0.5 cm3 of 100-proof whiskey per pound of body weight.

What is the effect of alcohol on useful consciousness? H0: D = No alcohol - Alcohol = 0 vs. H1: D > 0 (one sided) Paired T for NoAlcohol - Alcohol N Mean StDev SE Mean NoAlcohol 10 546.6 238.8 75.5 Alcohol 10 351.0 210.9 66.7 Difference 10 195.6 230.5 72.9 95% CI for mean difference: (30.7, 360.5) T-Test of mean difference = 0 (vs > 0): T-Value = 2.68 P-Value = 0.013

What is the effect of time on memory recall? 8 people were given 10 minutes to memorize a list of 20 nonsense words. Each was asked to list as many words as he or she could remember after 1 hour and again after 24 hours.

What is the effect of time on memory recall? Paired T for 1hour - 24hour N Mean StDev SE Mean 1hour 8 12.75 3.69 1.31 24hour 8 9.13 3.52 1.25 Difference 8 3.625 2.066 0.730 95% CI for mean difference: (1.897, 5.353) T-Test of mean difference = 0 (vs not > 0): T-Value = 4.96 P-Value = 0.001