Statistic of the Week

Inferential Statistics T-Tests

7400.685.080 Research Methods in HE/FE Inst: D. Witt

A T-Test is a test for Significant Differences between two means

Remember when we discussed "Descriptive Statistics" (means & standard deviations, modes and medians). These are calculated numbers that give us (at a glance) a picture of the data we are dealing with. These statistics - means and standard deviations - give us a summary of the raw data.

Inferential Statistics take us beyond the raw data - to generalizations from small samples to entire populations.

For example: Let's suppose Senator Phoggblatt wants to know how much his recent dirty dealings with dubious debtors has hurt his chance for re-election to a third term in office.

By comparing public opinion polls taken before the news of his indictment to polls taken after the news hit the press, the Senator can assess the overall damage to his reputation.

Even though the REAL TEST of the effects of his unscrupulous behavior will come at election time, the Senator can make a decision about whether or not he SHOULD run again, avoiding any of those unpleasant campaign debts..

One way to assess the damage is to run a T-TEST on the data. This is a common way to determine whether two or more groups are meaningfully different on some variable, issue, behavior, or attitude. A T-Test is simply a test of the statistical difference between two means - in this case the average popularity of the Senator before and after he was found out.

Here's a researchable question

Is the Eastern U.S. more violent than the West?
Yes or No?

We all have our biases, but a T-Test can help answer the question,
if we have viable indicators of Violence to compare.

We decide to let the Homicide Rates for 36 Eastern and 36 Western metropolitan areas be a measure of violence (an index of violence, really). By calculating means and standard deviations for Eastern Cities and Western Cities, we can statistically compare the means - and get an answer!

Here's the Data:

East Homicides Rate

West Homicides Rate

Mean Homicides



Std. Deviation



Our hypotheses is that the East is more Violent than the West.

Graphing the normal distribution for East and West would give us a bell shaped curve.

95% of either sample would fall within a plus or minus 2 Std. Deviations of its mean.

In other words:
A 95% Confidence Interval on each would give a range of:
7.91 to 11.59 for the East and 7.23 to 10.23 for the West.

This just means that the range from the East's low of 7.91
and the West's High of 10.23 is common to both geographic areas.
It also means that the average for either East or West will have to fall outside this range.

Are the two areas significantly different in terms of violence?

To find out, we perform a t-test, calculating the t statistic with the following formula:

The little e's and w's just mean east and west!

By substitution, in this case:

t = (9.75 - 8.73) / SQRT((5.67/36) + (4.19/36)) = 1.930
------------------(sqrt is an abbreviation for square root)

This is the T-Value (obtained)
and is the statistical difference between means divided by the standard error of the difference.

To see if this t-value is SIGNIFICANT or not, we need two other bits of information.

- Degrees of Freedom
in this case df = (Ne -1) + (Nw - 1) = (36 - 1)+(36 - 1) = 70

-A Distribution of t Table (like the Chi2 Distribution) to get the comparable Critical t-value.


  1. Simply locate the degrees of freedom (df) down the left hand side of the t-table.
  2. Run your finger across the values until you get to a value greater than your t-value obtained.
  3. The value just to the left of this greater than obtained value is in the column that shows your level of significant difference.

Click here for T-Table

t-value obtained is 1.930
critical t-value is 1.658 at 120 degrees of freedom.
Significance level is p<.10 for a two-tailed test -or p<.05 for a one-tailed test

Since the t-value obtained is bigger than the critical t-value at p,.05,
we conclude that there is a significant difference between East and West.
The East is more violent than the West, when we use Homicide rates as the measure of violence.

We use the one tail test because we are looking for a more violent area, not simply a difference..

Name ______________________________________________________ Homework Assignment: T-Tests

Now it is your turn:

1. This is a test of the "Just Who is Scatterbrained" Hypothesis that asserts that men can't handle multiple role expectations in their lives. When they cook a meal, they attend to the food and that's it - no being a parent in the middle of cooking, no phone conversations while preparing food. On the other hand, women seem to be able to be more than one person at a time.

You are a researcher interested in how multiple role expectations (MREs) affect men vs. women.

You survey men and women and find a sample of each (44 men and 51 women) who normally experience a similar level of MRE and you measure their level of self-esteem (the higher the score, the more esteem).

Your Data show: Women Men
Average Self-Esteem 2.56 2.39
Std. Deviation 0.52 0.54

Is the difference between men and women a statistically significant different? Do a t-test to find out.

Next, find a journal article where the authors are trying to judge significant differences between two sub-populations. Locate sample sizes, means and std. deviations for their data and perform your own t-test to see if the authors were correct in their interpretation of the data.

32. Now this one is a little harder, but I'll give you everything you need to answer all the parts.

Suppose you are trying to discover whether or not college men in fraternities spend more money on clothes than college men who are not in fraternities (independents).

Frat Boys spend an average of $765 on clothing each year, with a standard deviation of $44.34

Indie Boys spend an average of $700 on clothing each year, with a standard deviation of $33.23

There are 500 Frat Boys and 352 Indie Boys in your sample.

a. State a hypothesis that would suggest Frat Boys spend significantly more money on clothes.

b. Write out the appropriate formula for calculating a t and show the necessary calculations.

c. Specify the Degrees of Freedom (df) and the Cricital t.

d. Indicate whether the difference is significant or not and specify the level of significance.

e. Based on your calculations, Reject or do not Reject your hypothesis and interpret your findings.

Click here for T-Table