Hypothesis Testing with Two Samples

Read this chapter, which discusses how to compare data from two similar groups. This is useful when, for example, you want to analyze things like how someone's income relates to another sample that you are interested in. Make sure you read the introduction as well as sections 10.1 through 10.6. Attempt the practice problems and homework at the end of the chapter.

Cohen's Standards for Small, Medium, and Large Effect Sizes

Cohen's d is a measure of "effect size" based on the differences between two means. Cohen's d, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen's standards of small, medium, and large effect sizes.

Size of effect	d
Small	0.2
Medium	0.5
Large	0.8

Table 10.2 Cohen's Standard Effect Sizes

Cohen's d is the measure of the difference between two means divided by the pooled standard deviation:

$d=\dfrac{\overline x_1– \overline x_2}{s_{pooled}}$
where

$s_{pooled}=\sqrt{\dfrac{(n_1–1)s^2_1+(n_2–1)s^2_2}{n_1+n_2^{–2}}}$

It is important to note that Cohen's d does not provide a level of confidence as to the magnitude of the size of the effect comparable to the other tests of hypothesis we have studied. The sizes of the effects are simply indicative.

Example 10.4

Problem
Calculate Cohen's d for example 10.2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.

Solution 1

$\overline x_1 = 4 s_1 = 1.5 n_1 = 11$

$\overline x_2 = 3.5 s_2 = 1 n_2 = 9$

$d = 0.384$

The effect is small because 0.384 is between Cohen's value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences of the means for the two companies is small indicating that there is not a significant difference between them.