Significance Test: Paired t- Test (dependent t- Test) Dr. Amjad El-Shanti MD, PMH,Dr PH University of Palestine 2016

Paired t-Test The paired t- test is used to test whether the difference between a pair of variables measured on each individual is on the average zero.

Dependent Samples t Formulas: t is the difference in means over a standard error. We are still dealing with the Sampling Distribution of the Difference between the means. Our subscript is different here, but says basically the same thing. We are looking at the MEAN DIFFERENCE SCORE. The subscript for the independent samples t said we were looking at the DIFFERENCE BETWEEN THE MEANS. The standard error is found by finding the difference between each pair of observations. The standard deviation of these difference is SDD. Divide SDD by sqrt(number of pairs) to get SEdiff.

Another way to write the formula In this formula, we just put the formula for Sediff in the denominator instead of having you calculate it separately. [this is the formula that appears on the “Guide to Statistics” sheet they can download. In this formula, we just put the formula for Sediff in the denominator instead of having you calculate it separately.

D= ∑ Di n SDD = ∑Di2 – (∑Di)2 √ n-1

Distribution of the Paired t-Statistic Suppose x is a variable on each of two populations whose members can be paired. Further suppose that the paired-difference variable D is normally distributed. Then, for paired samples of size n, the variable has the t-distribution with df = n – 1. The normal null hypothesis is that μD = 0

The paired t-test for two population means Step 1 The null hypothesis is H0: D = 0; the alternative hypothesis is one of the following: Ha: D  0 Ha: D < 0 Ha: D > 0 (Two Tailed) (Left Tailed) (Right Tailed) Step 2 Decide on the significance level,  Step 3 The critical values are ±t/2 -t +t with df = n - 1.

The paired t-test for two population means

The paired t-test for two population means Step 4 Compute the value of the test statistic Step 5 If the value of the test statistic falls in the rejection region, reject H0, otherwise do not reject H0.

Example: The number of doses of medication needed for asthma attacks before and after relaxation training.

Step 1: Ho : there is no significant difference in the average of dose of medication in the week before relaxation training and in the week after relaxation training X (medication Dose)before = X (medication Dose)after Ha: there is significant difference in the average of dose of medication in the week before relaxation training and in the week after relaxation training Step 2: α =0.05

Decision Step 3:Critical t (df= 5-1=4, α=0.05) = 2.78 Step 4: Test t or (Observed t)= 3.72 Test t > Critical t P-value < α P-Value <0.05 Reject Ho , So can not reject Ha Decision: There is significant difference between the doses of medication in the week before training of relaxation and in the week after training of relaxation