CK-12 Probability and Statistics - Advanced

8.4. Testing a Hypothesis for Dependent and Independent Samples http://www.ck12.org

where:

s^2 d=sample variance

d=difference between corresponding pairs within the sample

d ̄=the difference between the means of the two samples

n=the number in the sample

sd=standard deviation

With the standard deviation, we can calculate thestandard errorusing the following formula:

sd ̄=

sd

√

n

After we calculate the standard error, we can use the general formula for the test statistic:

t=

d ̄−δ

sd ̄

This may seem a bit confusing, but let’s take a look at an example to help clarify.

Example:

The math teacher wants to determine the effectiveness of her statistics lesson and gives a pre-test and a post-test to
9 students in her class. Our hypothesis is that there is no difference between the means of the two samples and our
alternative hypothesis is that the two means of the samples are not equal. In other words, we are testing whether or
not these two samples are related or:

H 0 :δ=μ 1 −μ 2 = 0

H 0 :δ=μ 1 −μ 26 = 0

The results for the pre- and post-tests are below:

TABLE8.9:

Subject Pre-test Score Post-test Score d=difference d^2

1 78 80 2 4

2 67 69 2 4

3 56 70 14 196

4 78 79 1 1

5 96 96 0 0

6 82 84 2 4

7 84 88 4 16

8 90 92 2 4

9 87 92 5 25

Sum 718 750 32 254

Mean 79. 7 83. 3 3. 6

Using the information from the table above, we can first solve for the standard deviation of the two samples, then
the standard error of the two samples and finally the test statistic.