Objective: To test claims about inferences for two sample means from matched-pair tests or blocked samples, under specific conditions

8.3: PAIRED T-TEST

• Data are paired when the observations are collected in pairs or the observations in one group are naturally related to observations in the other group.

• Paired data arise in a number of ways. Perhaps the most common is to compare subjects with themselves before and after a treatment.

• When pairs arise from an experiment, the pairing is a type of blocking.

• When they arise from an observational study, it is a form of matching.

PAIRED DATA

• If you know the data are paired, you can (and must!) take advantage of it. • To decide if the data are paired, consider how they

were collected and what they mean.• There is no test to determine whether the data are

paired.

• Once we know the data are paired, we can examine the pairwise differences. • Because it is the differences we care about, we

treat them as if they were the data and ignore the original two sets of data.

PAIRED DATA (CONT.)

• Now that we have only one set of data (the differences) to consider, we can return to the simple one-sample t-test.

• Mechanically, a paired t-test is just a one-sample t-test for the means of the pairwise differences.

• The sample size is the number of pairs.

PAIRED DATA (CONT.)

• Independence Assumption• Independence Assumption: The differences

must be independent of each other. • Randomization Condition: Were the data

collected with suitable randomization (representative random samples or a randomized experiment)?

• 10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

PAIRED T-TEST ASSUMPTIONS &

CONDITIONS

• Normal Population Assumption: We need to assume that the population of differences follows a Normal model.

• Nearly Normal Condition: Check this with a histogram or Normal probability plot of the differences.

• Paired Data Assumption: The data must be paired.

PAIRED T-TEST ASSUMPTIONS &

CONDITIONS (CONT.)

• When the conditions are met, we are ready to test whether the paired differences differ significantly from zero.

• We test the hypothesis H0: d = 0, where the d’s are the pairwise differences and 0 is almost always 0.

THE PAIRED T-TEST

• We use the statistic

where n is the number of pairs.

is the ordinary standard error for the

mean applied to the differences.

• When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t-model on n – 1 degrees of freedom, so we can use that model to obtain a P-value.

THE PAIRED T-TEST (CONT.)

0

1nd

tSE d

dsSE dn

1. Check Conditions and show that you have checked these!• Independence Assumption: The differences must

be independent of each other. • Randomization Condition: Were the data

collected with suitable randomization (representative random samples or a randomized experiment)?

• 10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

• Nearly Normal Condition: Check this with a histogram or Normal probability plot of the differences.

• Paired Data Assumption: The data must be paired.

STEPS FOR PAIRED T-TEST

2. State the test you are about to conduct Ex) Paired t-Test

3. Set up your hypothesesH0:HA:

4. Calculate your test statistic

5. Draw a picture of your desired area under the t-model, and calculate your P-value.

STEPS FOR PAIRED T-TEST (CONT.)

6. Make your conclusion.

PAIRED T-TEST (CONT.)

P-Value Action ConclusionLow Reject H0 The sample

mean is sufficient evidence to conclude HA in context.

High Fail to reject H0

The sample mean does not provide us with sufficient evidence to conclude HA in context.

We hear that listening to Mozart improves students’ performances on tests. To test this idea, 21 subjects worked a paper-and-pencil maze while wearing a mask. The mask was either unscented or carried a floral scent. The response variable is their average time on three trials. Each subject worked the maze with both masks, in a random order. The randomization is important because subjects tend to improve their times as they work a make repeatedly. The table below gives the subjects’ average times with both masks.

PAIRED T-TEST EXAMPLE

PAIRED T-TEST EXAMPLE (CONT.)

• When the conditions are met, we are ready to find the confidence interval for the mean of the paired differences.

• The confidence interval is

where the standard error of the mean difference is

The critical value t* depends on the particular confidence level, C, that you specify and on the degrees of freedom, n – 1, which is based on the number of pairs, n.

PAIRED T-INTERVAL

1nd t SE d

dsSE dn

1. Check Conditions and show that you have checked these!• Independence Assumption: The differences must

be independent of each other. • Randomization Condition: Were the data

collected with suitable randomization (representative random samples or a randomized experiment)?

• 10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

• Nearly Normal Condition: Check this with a histogram or Normal probability plot of the differences.

• Paired Data Assumption: The data must be paired.

STEPS FOR PAIRED T-INTERVAL

2. State the test you are about to conduct Ex) Paired t-Interval

4. Calculate your t-interval

5. State your conclusion IN CONTEXT.I am 95% confident that 6 weeks of this exercise program

can produce a mean decrease in resting pulse rate of between 1 and 4.6 beats per minute.

STEPS FOR PAIRED T-INTERVAL (CONT.)

A study is conducted on 11 people diagnosed as being dependent on caffeine. Each subject was barred from coffee, colas, and other substances containing caffeine. During a different time period, they took placebo capsules. The order in which subjects took caffeine and the placebo was randomized. The table below contains data on two of several tests given to the subjects. “Depression” is the score on the Beck Depression Inventory. Higher scores show more symptoms of depression. “Beats” is the beats per minute the subject achieved when asked to press a button 200 times as quickly as possible. We are interested in whether being deprived of caffeine affects these outcomes. Conduct a 90% confidence interval for the mean change in depression score.

PAIRED T-INTERVAL EXAMPLE

PAIRED T-INTERVAL EXAMPLE (CONT.)

• In 8.1-8.2, the data came from two different and independent groups, so we used a two-sample t-test.

• Now the data are two different measurements for the same group or two groups in which individuals were paired, so we will use a paired t-test.

RECAP

• Day 1: 8.3 Book Page # 1, 20, 22, 24

• Day 2: 8.3 Book Page # 5, 6, 16, 25, 26

ASSIGNMENTS