Testing Hypotheses for Means

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $$ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $$ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $$$ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $$$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ The t Test for Related Samples The t Test for Related$Samples Program Transcript MATT JONES: As$its$name implies, the independent samples$t E test has$the assumption of the independence of observations. But that's not always$the case.

Sometimes$we take multiple observations$of the same unit of analysis, such as$a person, over$time. In th is$case, we'll use a paired sample t E test, sometimes$ referred to as$the dependent sample t E test. Let's go to SPSS$to see how we do this.

To perform$the paired sample t E test in SPSS, we once again go to Analyze, Compare Means, and down to the Paired Sample T E test. SPSS$doesn't require much information hereR only$the pair$of variables$of which we would like to test.

We have a simulated data set here for$statistical anxiety$of students. Students$ were provided with an instrument that measures$their$anxiety$aro und statistical topics$on a number$of different constructs EE teachers, interpretation, asking for$ help, worth, and self E conceptualization.

They$were given the test at the beginning of the class$and at the conclusion of a class. Hence, why$in the value lab els$we see pre E test and post E test. As$a teacher, I might have some interest in determining whether$students$felt more comfortable with me or$had lowering anxiety$over$time. This$is$perfect for$a paired sample t E test. To perform$this$paired sample t E test, w e'll go to Analyze, Compare Means, the Paired Sample T E test.

SPSS doesn't ask$for$much informationR only$the pair$of variables$of which I would like to test. In this$case, teacher$pre E test and teacher$post E test. So this$is$a classic$before and after. The first piece of output I obtain from$the paired sample t E test are some descriptive statistics, specifically$around the pairwise comparison I'm looking at, which is$the teacher$subscale pre E test and post E test.

I see that there is$mean on the pre E test of 17.

32 and on the post E test, an 18.44.

So it appears, at least from$a descriptive sense, that there is$a higher$mean on the post E test than the pre E test. On the instrument, higher$scores$on an item$or$the subscale indicate higher$levels$of anxiety$for$that spec ific$attitude. Except for$this$ specific$subscale, fear$of statistics$teachers, where higher$scores$actually$ indicate lower$levels$of anxiety.

So if post scores$are higher$than pre scores, that means$on average, students$ feel lower$levels$of anxiety$and mo re positive attitude about their$statistics$ teacher. I can see here, at least from$a descriptive sense, that that appears$to be the case. But from$the sample, I am$performing a test of statistical significance.

Next to the mean, I'm provided with the sampl e size 25 EE 25 observations$pre E test and 25 observations$post E test, all the same person EE the standard deviation for$ the pre E test and the post E test, and the standard error$of the mean.

1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $$ $ $$ $$ $ $ $ $ $ $ $ $$ $ $$ $ $$ $ $ $$ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ $ $ The t Test for Related Samples Next, let's go down and interpret the paired sample test itself. We ca n see that on average, there was$a difference of 1.12 units$on the scale with a standard deviation of 2.50. From$the 95% confidence interval, we see that the true difference is$somewhere between 2.15 and 0.085. We have a t E statistic$of 2.235 and an associa ted p E value of 0.035.

At the 0.05 level, the results$are statistically$significant and we can say$that there is$a significant difference between pre E test scores$and post E test scores.

Therefore, we can reject the null hypothesis$that there is$no difference . On average, it appears$on the post E test, students$had lower$levels$of anxiety$about their$statistics$teacher.

This$last example illustrated that students$felt more comfortable with statistics$as$ time progressed and specifically$felt less$anxious$about t heir$statistics$instructor. I certainly$hope this$example rings$true for$you, and that you feel comfortable or$at least don't self E identify$as$being anxious$about statistics$at the conclusion of this$ course. I encourage you to review your$textbook, review the videos, ask$your$ instructor$for$help, and also research the resources$here available at Walden University$to help you succeed.