CK-12 Probability and Statistics - Advanced

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

Dependent samplesare a bit different. Two samples of data are dependent when each score in one sample is paired
with a specific score in the other sample. In short, these types of samples are related to each other. Dependent
samples can occur in two scenarios:

• A group may be measured twice such as in a pretest-posttest situation (scores on a test before and after the
  lesson).


• In a matched sample where each observation is matched with an observation in the other sample.

To distinguish between tests of hypotheses for independent and dependent samples, we use a different symbol for
hypotheses with dependent samples. For dependent sample hypotheses, we use the delta symbol(δ)to symbolize
the difference between the two samples. Therefore, in our null hypothesis we state that the difference of scores
across the two measurements is equal to 0(δ) =0 or:

H 0 :δ=μ 1 −μ 2 = 0

Calculating the Pooled Estimate of Population Variance

When testing a hypothesis about two independent samples, we follow a similar process as when testing one random
sample. However, when computing the test statistic, we need to calculate the estimated standard error of the
difference between sample means(sX ̄ 1 −sX ̄ 2 ). Usually, with one sample this calculation is pretty easy since it is
based on either standard deviation of the sample or the population variance. However, when calculating this statistic
for two samples, it is a bit more difficult. To calculate this statistic we use the formula:

sX ̄ 1 −X ̄ 2 =

√

s^2

(

1

n 1

+

1

n 2

)

,

Wheren 1 and n 2 the sizes of the two samples

s^2 =the pooled sample variance, which is computed as shown below

The pooled estimate of variance is found by adding the sums of the squared deviations(s)around the sample means
and then dividing the total by the sum of the degrees of freedom in the two samples.

Therefore, we can find this estimate by using the formula:

s^2 =∑

(X 1 −X ̄ 1 )^2 +∑(X 2 −X ̄ 2 )^2

n 1 +n 2 − 2

Often, the top part of this formula is simplified by substituting the symbol SS for the sum of the squared deviations.
Therefore, the formula often is expressed by:

s^2 =

SS 1 +SS 2

n 1 +n 2 − 2

Let’s calculate this estimate using a sample set of data.

Example:

Say that we have two independent samples of student reading scores. The data are as follows: