Two-Sample T-Tests: Comparing Means of Two Independent Groups, Lecture notes of Economic statistics

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Two-sample t-tests.

• Independent samples
• Pooled standard devation
• The equal variance assumption

Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular value.

One-sided test:

Two-sided test:

In every one sample test, we have a given value we’re comparing the sample mean against. The question: Is this given value plausible?

But what if we don’t have a specific value to compare against?

What if, instead, we’re comparing the means of two groups

against each other?

That’s a job for two-sample testing.

The alternative can be two-sided (not-equal), or one-sided on either side (less than / more than)

Another way to say “two means at the same” is

“The difference between these means is zero .”

Let 1 and 2 be the means of each of the samples.

Let 1 and 2 be the true means, (that H 0 says are the same)

The t-score looks like:

We never deal with the formula in this form. Since the null assumption is that the true means are the same.

A hairy topic, to be sure. How about an example?

Example: New textbook vs. old textbook.

Say we wanted to test if a new brand of textbook is a better resource as measured by provincial exam scores. (alpha = .05)

We’ll get two classrooms of the same grade and similar skill level. We’ll then flip a coin, if heads:

Class A gets the current textbook, Class B get the new one.

Otherwise, Class A gets the new one.

First, identify:

We are comparing two mean, a two-sample test is appropriate.

We want to know if the new textbook is better so this is a one- sided test.

But what do we do about the standard error?

There are values for n, and two values for standard deviation.

Old text class New text class Mean 64.3 68. Std. Dev. s

Sample Size n 21 23

We can get a measure that collects ( pools ) the standard

deviation of the whole system.

It’s called the pooled standard deviation.

In this case, the samples are roughly the same size, so the

pooled standard deviation SP should be near the middle of

the two standard deviations (7.1 and 7.4).

The degrees of freedom of the pooled standard deviation SP is

the total of the df from each sample. (n 1 – 1) + (n 2 – 1)

SP =7.26 is used to find the standard error.