STATS FOR FINAL RESEARCH PROJECTS - T-TEST

1. First, find the grand mean (X) for each treatment:

X = (xi)/ni

xi is the mean of each replicate and ni is the number of replicates in that treatment. If you only have one observation for each replicate, then this one number is considered to be xi. You should have one grand mean (X) for each of your treatments.

2. Now find the variance (s2) for each treatment:

s2 = (xi 2)- [(xi) 2/ni]

______

(ni -1)

xi is the mean of each replicate and ni is the number of replicates in that treatment. You should have a different variance for each treatment. For example, if you have three treatments then you should calculate three separate variances, one for each treatment.

2. The standard error (SE) can be calculated for each treatment. First take the square root of the variance (s2) of each treatment to find the standard deviation (s). Then divide the standard deviation by the square root of the number of replicates you have for each treatment. You should have a standard error for each of your treatments. The standard error can be used for your graphs (plot the mean for each treatment  SE). You do not need to use the standard error for your t-test, but you will be required to report it in a table in your final report. The formula for the standard error is below:

SE = s/ ni

4. Now you must perform a t-test for each pairwise comparison. For example, if you are comparing three treatments, you will have to perform three pairwise comparisons and therefore three separate t-tests (Treatment 1 vs. Treatment 2, Treatment 2 vs. Treatment 3, and Treatment 1 vs. Treatment 3) If you are comparing four treatments, you will have to perform six pairwise comparisons and therefore six separate t-tests (Trt 1 vs. Trt 2, Trt 2 vs. Trt 3, Trt 3 vs. Trt 4, Trt 1 vs. Trt 3, Trt 1 vs. Trt 4, Trt 2 vs. Trt 4).

Pick the first pair of comparisons and find Sp ,the pooled standard deviation for that particular pair. You will have a separate pooled standard deviation for each pair of comparisons.

Sp = (n1 -1) s12 + (n2 -1) s22

______

(n1 + n2 -2)

n1 and n2 are the number of replicates for the first and second treatment in your pairwise comparison. s12 and s22 are the variances of the mean for the first and second treatment in each pair (calculated in #2 above).

Next take the numbers found in the above steps and plug into the formula for a t-test.

t = X1 - X2

______

Sp 1/n1 + 1/n2

X1 and X2 are the treatment grand means (calculated in #1 above) for each treatment in our pairwise comparison. Make sure that X1 is greater than X2 to avoid getting negative numbers.

5. You should have a separate t-test and therefore a separate t-value for each of your pairwise comparisons. Now you must compare your calculated t-value to a table value. The degrees of freedom (df) for each test is defined as n1 + n2 - 2. If all of your number of replicates are the same, then the degrees of freedom for each t-test are the same. If your number of replicates for each treatment is different, you will have a different degrees of freedom for each t-test. The overall  value for your tests will be 0.10. Now go to the t-table. Go down the first column to find appropriate number of degrees of freedom (df). If the exact value for your df is not in the table, use the closest number to your actual df that is not greater than your actual df. For example, if you have 50 df, you would use the table value corresponding to 40 df, not 60 df. Next go across the table until 1 minus /2 (0.95) is on the top column. This value is the table t-value that you will compare your calculated t-value to. If your calculated t-value is greater than the table value, you reject your null hypothesis and conclude that there is a difference between the two treatments. If your calculated t-value is less than the table t-value, you accept your null hypothesis and conclude that there is no difference between treatments.

PRESENTATION OF DATA FOR FINAL PAPER AND TALKS:

Include a table containing the following: means for each treatment, standard error for each treatment, number of replicates for each treatment and a summary of each t-test performed. For the summary of the t-test, first identify the pairwise comparison, then report the df, calculated t-value, adjusted  value, and table t-value. The table below is an example of how you might present the above data.

TreatmentMeanSENumber of Replicates

pH 3.05.21.03

pH 6.07.50.93

pH 9.06.23.52

Summary of t-testsdft-valuetable t-value (=.015)* or NS

pH 3.0 vs. pH 6.044.53.298*

pH 6.0 vs. pH 9.033.93.896*

pH 3.0 vs. pH 9.042.23.298NS

adjusted  = .0167

In order to graphically display your data, plot the means  SE of each treatment and indicate which differed from one another when you performed your t-tests. Use letters to designate which treatments differed when you used the t-tests. In the graph, the treatments labeled with the same letters are not different from one another. In the sample graph below, the pH 3.0 treatment differed from the pH 6.0 treatment, so one is labeled “a” and the other is labeled “b,” thus indicating the difference. The pH 9.0 treatment was not different from either the pH 3.0 or pH 6.0 treatment and is labeled with “ab.” This notation indicates that the pH 9.0 treatment (ab) was not different from the pH 6.0 treatment (a) or the pH 3.0 treatment (b) since it is labeled with letters in common with both treatments.

DO NOT INCLUDE RAW DATA IN TALKS OR PAPERS!! Don’t forget to include descriptive titles for all tables and graphs.