![]() The practical conclusion made by the instructor is that the tests are not of equal difficulty. You reject the hypothesis that the mean difference is zero. The test statistic is higher than the t value.Next year, she can use both exams and give half the students one exam and half the other exam. The practical conclusion made by the instructor is that the two tests are equally difficult. You fail to reject the hypothesis that the mean difference is zero. The test statistic is lower than the t value.There are two possible results from our comparison: The t value with α = 0.05 and 15 degrees of freedom is 2.131. Statisticians write the t value with α = 0.05 and 15 degrees of freedom as: The degrees of freedom ( df) are based on the sample size and are calculated as: In our exam score data example, we set α = 0.05. We compare the test statistic to a t value with our chosen alpha value and the degrees of freedom for our data. Next, we calculate the standard error for the score difference. Software will usually display more decimal places and use them in calculations.) (Note that the statistics are rounded to two decimal places below. To accomplish this, we need the average difference, the standard deviation of the difference and the sample size. We start by calculating our test statistic. We'll further explain the principles underlying the paired t-test in the Statistical Details section below, but let's first proceed through the steps from beginning to end. We test if the mean difference is zero or not. We calculate the difference in exam scores for each student. The instructor wants to know if the two exams are equally difficult. An instructor gives students an exam and the next day gives students a different exam on the same material.We want to know if the mean weight change for people in the program is zero or not. For each person, we have the weight at the start and end of the program. We measure weights of people in a program to quit smoking.Then we test if the mean difference is zero or not. Since we have pairs of measurements for each person, we find the differences. We do this by finding out if the arm with medicated lotion has less redness than the other arm. We want to know if the medicated lotion is better than the non-medicated lotion. After a week, a doctor measures the redness on each arm. A group of people with dry skin use a medicated lotion on one arm and a non-medicated lotion on their other arm.We also have an idea, or hypothesis, that the differences between pairs is zero. Other times, we have separate variables for “before” and “after” measurements for each pair and need to calculate the differences. Sometimes, we already have the paired differences for the measurement variable. One variable defines the pairs for the observations. For the paired t-test, we need two variables.
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