Statistical Analysis

Resource Sheet pdf

How to Statistically Analyze Experimental Data

Step One: Identify null hypothesis and alternative hypothesis. (Researcher should have set up both hypotheses before beginning experiment.)

· Null Hypothesis: The experimenter assumes there is no effect of the treatment or no relationship between two variables.

· Alternative Hypothesis: What the experimenter thinks may be true or wishes to be true before beginning the experiment.

Step Two: Calculate the means for both samples.

Step Three: Calculate the variance for both samples. For all calculations round the decimal to the nearest thousandth.

· First: Calculate the deviation (or difference of the data point from the mean) for each data point.

· Second: Square these deviations to ensure that all values are positive.

· Third: Calculate the sum of all of these squared deviations.

· Fourth: Divide by the number of data points in the data set minus one.

Step Four: Calculate the standard deviation for both samples.

· Take the square root of the variance.

Step Five: Determine which type of t-test is most appropriate (student’s t-test vs. paired t-test).

· Student’s t-test: Used to determine whether the difference between the means of two independent groups (both of which are being tested for the same dependent variable) is statistically significant.

· Paired t-test: Used to determine whether the difference between the means of two groups (each containing the same participants and being tested at two different points) is statistically significant.

· If the student’s t-test is most appropriate, determine which formula variation is most appropriate.

Step Six: Use the t-test formula to calculate the t value.

· If completing a paired t-test, you will also need to calculate the Difference Score (D) for each participant in the study group. See Example 3 on the Example T-Test Calculations Resource Sheet.

D = Patient 1’s mean before treatment – Patient 1’s mean after treatment

Step Seven: Calculate the degrees of freedom.

Step Eight: Determine whether you completed a one-tailed or two-tailed test for significance.

· One-tailed test if alternative hypothesis was directional.

· Two-tailed test if alternative hypothesis was nondirectional.

Step Nine: Use the t-table to compare the t value with the critical values to determine whether the results are statistically significant.

Determine whether the t value exceeds any of the critical values in the corresponding row.

· If the t value exceeds any of the critical values in the row, the alternative hypothesis can be accepted. This means that the results ARE STATISTICALLY SIGNIFICANT.

· If the t value is smaller than all of the critical values in the row, the alternative hypothesis is rejected and the null hypothesis can be accepted. This means that the results are NOT STATISTICALLY SIGNIFICANT.

If the t value exceeds any of the critical values in the row, you now need to:

· Follow the corresponding row over to the right until you find the column with the critical value that is just slightly smaller than the t value.

· Follow this column to the top of the table to determine its corresponding p value.

· Determine which p value to use: the p value corresponding with a one-tailed test for significance versus the p value corresponding with a two-tailed test for significance.

The p values indicate the probability that the difference between the means of the two samples is due only to chance.

· If the p value ≤ 0.01, the results are VERY SIGNIFICANT.

o The probability that the difference is due to chance is less than or equal to 1%.

· If the p value ≤ 0.05, the results are SIGNIFICANT.

o The probability that the difference is due to chance is less than or equal to 5%.

· If the p value > 0.05, the results are NOT SIGNIFICANT.

o The probability that the difference is due to chance is greater than 5%.

Step Ten: Reject or accept null hypothesis.

Step One: Identify null hypothesis and alternative hypothesis.

· Null hypothesis: Females develop the same number of colds as males.

· Alternative hypothesis: Males develop fewer colds than females.

Step Two: Calculate the means for both samples.

Step Three: Calculate the variance for both samples.

Calculate the deviation for each data point.

Square these deviations.

Calculate the sum of all of these squared deviations.

Divide by the number of data points in the data set minus one.

Step Four: Calculate the standard deviation for both samples.

· Take the square root of the variance.

Step Five: Determine which type of t-test is most appropriate (student’s t-test versus paired t-test).

The samples are independent of each other (i.e., different subjects were used in the two samples). Therefore, a student’s t-test is the most appropriate t-test to use. The two samples are unequal AND both sample sizes are small (<30), therefore the following formula should be used:

Step Six: Use the t-test formula to calculate the t value.

Step Seven: Calculate the degrees of freedom.

Step Eight: Determine whether you completed a one-tailed or two-tailed test for significance.

The alternative hypothesis, “Males develop fewer colds than females” is directional. Therefore you completed a one-tailed test for significance.

Step Nine: Use the t-table to compare the computed t value with the critical values to determine whether the results are statistically significant.

Because the t value is less than all of the critical values in the t table, we can accept the null hypothesis that there is no difference between the numbers of colds developed in males versus females. The difference between the means is not statistically significant.

Example Two:

Step One: Identify null hypothesis and alternative hypothesis.

· Null hypothesis: The stress score of those participating in biofeedback therapy sessions will be the same as those not participating in biofeedback therapy sessions.

· Alternative hypothesis: The stress score of those participating in biofeedback therapy sessions will be less than those not participating in biofeedback therapy sessions.

Step Two: Calculate the means for both samples.

Step Three: Calculate the variance for both samples.

Calculate the deviation for each data point.

Square these deviations.

Calculate the sum of all of these squared deviations.

Divide by the number of data points in the data set minus one.

Step Four: Calculate the standard deviation for both samples.

· Take the square root of the variance.

Step Five: Determine which type of t-test is most appropriate (student’s t-test versus paired t-test).

The samples are independent (different subjects were used in sample 1 than in sample 2). Therefore, the student’s t-test is the most appropriate t-test to use. The two samples are equal; therefore, the following formula should be used:

Step Six: Use the t-test formula to calculate the t value.

Step Seven: Calculate the degrees of freedom.

Step Eight: Determine whether you completed a one-tailed or two-tailed test for significance.

The alternative hypothesis, “The stress score of those participating in biofeedback therapy sessions will be less than those not participating in biofeedback therapy sessions” is directional. Therefore you completed a one-tailed test for significance.

Step Nine: Use the t-table to compare the computed t value with the critical values to determine whether the results are statistically significant.

The t value of 4.034 is very significant, so we can reject the null hypothesis and accept the alternative hypothesis that states “The stress score of those participating in biofeedback therapy sessions will be less than those not participating in biofeedback therapy sessions.” The probability that this difference is due to chance is less than 0.005 (or 0.5%).

Example Three:

Step One: Identify null hypothesis and alternative hypothesis.

· Null hypothesis: There is no difference in the weights of the patients before and after treatment.

· Alternative hypothesis: The weights of the patients after treatment will be less than before treatment.

Step Two: Calculate the means for both samples.

Step Three: Calculate the variance for both samples.

Calculate the deviation for each data point.

Square these deviations.

Calculate the sum of all of these squared deviations.

Divide by the number of data points in the data set minus one.

Step Four: Calculate the standard deviation for both samples.

· Take the square root of the variance.

Step Five: Determine which type of t-test is most appropriate (student’s t-test versus paired t-test).

The samples are paired (the same subjects were used in sample 1 as in sample 2). Therefore, a paired t-test is the most appropriate t-test to use.

Step Six: Use the t-test formula to calculate the t value.

Step Seven: Calculate the degrees of freedom.

Step Eight: Determine whether you completed a one-tailed or two-tailed test for significance.

The alternative hypothesis, “The weights of the patients after treatment will be less than before treatment” is directional. Therefore you completed a one-tailed test for significance.

Step Nine: Use the t-table to compare the computed t value with the critical values to determine whether the results are statistically significant.

The t value of 5.109 is very significant, so we can reject the null hypothesis and accept the alternative hypothesis that states “The weights of the patients after treatment will be less than before treatment.” The probability that this difference is due to chance is less than 0.005 (or 0.5%).