Inferant Statistics

t-test Steps

Steps to perform a t-test

State the statistical hypotheses.
Null Hypothesis: The m up river site equals the m down river site
Alternate Hypothesis: The m up river site does not equal the m down river site
The null hypothesis states there is no difference between the population means the m and the alternate hypothesis states there is a difference between the population means.
Decide whether to use a normal t-test or a paired t-test.
Select the Alpha level.
Decide whether to do a one-tailed t-test or a two-tailed t-test.
Do the t-test using Excel or the TI-83 calculator (Input all data points not means).
Interpret the results and decide whether to:
Reject the Null Hypothesis and accept the Alternative Hypothesis
This is the case when the absolute value of the t-statistic > t-critical; and p<alpha.

or

Retain the Null Hypothesis and reject the Alternate Hypothesis
This is the case when the absolute value of the t-statistic < t-critical; and p>alpha)

In cases were the two groups being compared are not connected in any way, use the normal t-test. The case of the two groups of data collected at different sites on a river is an example where a normal t-test would be used. The data sets are independent of each other.

Next, consider the case where a group of students takes a pre-test, is given some form of instruction and then takes a post-test to see if the instruction was effective. The two sets of test data are connected since the same students provided both sets; each student has a pre and a post test! In this example a paired t-test would be used.

Alpha Levels

The alpha level you choose determines the risk you are willing to take of being wrong. Remember, if you reject the null hypothesis, it means the difference between the group means is so large it would rarely be found between two means from the same population...but it might be found! Setting the alpha level at 0.05 means you are willing to accept the risk of being wrong 5 times out of 100. Therefore, if you reject the null hypothesis, it means you could be wrong but the mean difference between the t-test groups would only be found between groups from the same population 5 times out of 100. Therefore, if you set the alpha level to 0.05, you'll be right 95 times out of 100! Are you willing to be correct 95 times out of 100? The most common alpha level used by researchers is 0.05. However, if you cannot afford to be wrong five times out of 100, set the alpha level to 0.01 and you risk being wrong only 1 time out of 100! Of course to reject the null hypothesis at the 0.01 alpha level, there will have to be a larger difference between the group means.

One-Tailed or Two-Tailed?

Generally, researchers select two-tailed tests because they are not sure whether the difference between the means being compared will be positive or negative. In cases where the researcher has done some previous research and is sure the difference between the means will be either positive or negative, a one-tailed test can be used. However, there is a greater chance of being wrong if a one-tailed test is used so the researcher really needs to know a lot about the data he is studying!

What's all this t-statistic, t-calculated and p stuff?

The t-critical is a number that represents the size of the mean difference required for the alpha level selected. The t-statistic is a number that represents the actual size of the difference between the two test means. Let's assume a=0.05. If t-statistic> t-critical is true, it means that the difference between the two test means is so large it would only be found 5 times out of 100 in means coming from the same population. Therefore, there is a 95% chance that the means come from different populations and the null hypothesis should be rejected. The p is the "calculated alpha." It represents the probability of the difference in the test means coming from the same population. Therefore, if p=0.04 (less than alpha), it means that the difference between the two test means is so large it would only be found 4 times out of 100 with means coming from the same population. Therefore, there is a 96% chance that the means come from different populations and the null hypothesis should be rejected. Conversely, what if p=0.10 (greater than alpha)? In this case, the difference between the two test means would be found 10 times out of 100 with means coming from the same population. Therefore, there is a 90% chance that the means come from different populations. However, there is a 10% chance that the means come from the same population! Since alpha =0.05, the 10% chance is greater than the 5% chance allowed and the null hypothesis should be retained.