In this tutorial we will discuss two sample t test for testing difference between two population means when the population variances are unknown and unequal.
Two sample t test for means with unknown and unequal variances
Let $\overline{x}_1$ be the sample mean and $s_1$ be the sample standard deviation of a random sample of size $n_1$ from a population with mean $\mu_1$ and variance $\sigma^2_1$.
Let $\overline{x}_2$ be the sample mean and $s_2$ be the sample standard deviation of a random sample of size $n_2$ from a population with mean $\mu_2$ and variance $\sigma^2_2$.
Suppose the variances $\sigma^2_1$ and $\sigma^2_2$ are unknown and unequal.
Assumptions
Assumptions for two sample $t$-test are as follows:
a. The population from which, the two samples drawn are Normal distributions.
b. The two population variances are unknown and unequal.
Step by Step Procedure
We wish to test the hypothesis $H_0 : \mu_1 = \mu_2$.
The standard error of difference between means is
$$ \begin{eqnarray*} SE(\overline{x}_1 - \overline{x}_2) &=& \sqrt{ \frac{s_1^2}{n_1}+ \frac{s_2^2}{n_2}}. \end{eqnarray*} $$
The step by step hypothesis testing procedure is as follows:
Step 1 State the hypothesis testing problem
The hypothesis testing problem can be structured in any one of the three situations as follows:
| Situation | Hypothesis Testing Problem |
|---|---|
| Situation A : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 < \mu_2$ (Left-tailed) |
| Situation B : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 > \mu_2$ (Right-tailed) |
| Situation C : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 \neq \mu_2$ (Two-tailed) |
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ \begin{eqnarray*} t & =& \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{SE(\overline{x}_1-\overline{x}_2)}\\\\ & =& \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}} \end{eqnarray*} $$
The test statistic $t$ follows Students' $t$ distribution with $\nu$ degrees of freedom, where
$$ \begin{eqnarray*} \nu & =& \frac{\bigg(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\bigg)^2}{\frac{s_1^4}{n_1^2(n_1-1)}+\frac{s_2^4}{n_2^2(n_2-1)}} \end{eqnarray*} $$
The value of $\nu$ can be rounded to the nearest integer.
Step 3 Specify the level of significance $\alpha$
Step 4 Determine the critical values
For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.
- For left-tailed alternative hypothesis: Find the $t$-critical value using
$$ \begin{aligned} P(t < -t_\alpha) = \alpha. \end{aligned} $$
- For right-tailed alternative hypothesis: $t_\alpha$.
$$ \begin{aligned} P(t > t_\alpha) = \alpha. \end{aligned} $$
- For two-tailed alternative hypothesis: $t_{\alpha/2}$.
$$ P(t < - t_{\alpha/2} \text{ or } t > t_{\alpha/2}) = \alpha. $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using equation
$$ \begin{eqnarray*} t_{obs} &= & \frac{(\overline{x}_1-\overline{x}_2)-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}} \end{eqnarray*} $$
Step 6 Decision (Traditional Approach)
It is based on the critical values.
- For left-tailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\leq -t_\alpha$. - For right-tailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\geq t_\alpha$. - For two-tailed alternative hypothesis: Reject $H_0$ if
$|t_{obs}|\geq t_{\alpha/2}$.
OR
Step 6 Decision ($p$-value Approach)
It is based on the $p$-value.
| Alternative Hypothesis | Type of Hypothesis | $p$-value |
|---|---|---|
| $H_a: \mu_1<\mu_2$ | Left-tailed | $p$-value $= P(t\leq t_{obs})$ |
| $H_a: \mu_1>\mu_2$ | Right-tailed | $p$-value $= P(t\geq t_{obs})$ |
| $H_a: \mu_1\neq \mu_2$ | Two-tailed | $p$-value $= 2P(t\geq abs(t_{obs}))$ |
If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.
Endnote
In this tutorial, you learned the $t$-test for testing two population means with unequal variances and the assumptions for $t$-test for testing two population means. You also learned about the step by step procedure to apply $t$-test for testing two population means with unequal variances.
To learn more about other hypothesis testing problems, hypothesis testing calculators and step by step procedure, please refer to the following tutorials:
Let me know in the comments if you have any questions on $t$-test for two means with unequal variances and your thought on this article.
