Testing Equality of Means of Two Normal Populations¶

Known Variances¶

Consider the two populations as

\[ \begin{aligned} X_{1}, X_{2}, \ldots, X_{n} &\sim \mathcal{N}(\mu_{x}, \sigma_{x}^{2})\newline Y_{1}, Y_{2}, \ldots, Y_{m} &\sim \mathcal{N}(\mu_{y}, \sigma_{y}^{2}) \end{aligned} \]

Then, the difference in the sample means will itself be a normal distribution (since it is the difference of two normals), and the test statistic can be defined as below

\[ \begin{aligned} \overline{X} - \overline{Y} \sim \mathcal{N}(\mu_{x} - \mu_{y}, \sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}})\newline T = \frac{(\overline{X} - \overline{Y}) - (\mu_{x} - \mu_{y})}{\sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}}} \sim \mathcal{N}(0, 1) \end{aligned} \]

when \(H_{0}\) is true,

\[ \begin{aligned} T = \frac{(\overline{X} - \overline{Y})}{\sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}}} \sim \mathcal{N}(0, 1) \end{aligned} \]

and we compare the following hypotheses

\[ \begin{alignat}{3} &H_{0}: \mu_{x} = \mu_{y} \quad &\text{versus} \quad &H_{1}: \mu_{x} \neq \mu_{y}\newline \text{or,} \quad &H_{0}: \mu_{x} - \mu_{y} = 0 \quad &\text{versus} \quad &H_{1}: \mu_{x} - \mu_{y} \neq 0 \end{alignat} \]

we reject \(H_{0}\) when the difference between the means is large, i.e., \(H_{0}\) is testing whether the test statistic is close to zero or not

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &T &> &z_{\alpha/2}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &T &\leq &z_{\alpha/2} \end{alignat} \]

where the variances of both the populations are known.

In a very similar fashion, the hypthesis testing rules for one sided test can be derived.

For

\[ \begin{aligned} H_{0}: \mu_{x} \leq \mu_{y} \quad \text{versus} \quad H_{1}: \mu_{x} > \mu_{y} \end{aligned} \]

We reject \(H_{0}\) when the difference \(\mu_{x} - \mu_{y}\) is highly positive

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &\frac{(\overline{X} - \overline{Y})}{\sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}}} &> &z_{\alpha}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &\frac{(\overline{X} - \overline{Y})}{\sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}}} &\leq &z_{\alpha} \end{alignat} \]

For the other side of the test, we use the same criteria as above, but switching the sets \(X\) and \(Y\).

Unknown but Equal Variances¶

We consider the same two populations of \(Xs\) and \(Ys\), but this time the variances are unknown. for simplicity of analysis, we assume that the unknown variances are same

\[ \begin{aligned} \sigma_{x} = \sigma_{y} = \sigma \end{aligned} \]

From section, we know

\[ \begin{aligned} S_{p}^{2} = \frac{(n-1)S_{1}^{2} + (m-1)S_{2}^{2}}{n + m - 2}\newline \frac{(\overline{X} - \overline{Y}) - (\mu_{x} - \mu_{y})}{\sqrt{\frac{\sigma_{x}^{2}}{n} + \frac{\sigma_{y}^{2}}{m}}} \div \frac{S_{p}}{\sigma} \sim t_{n+m-2} \end{aligned} \]

If \(H_{0}\) is true, \(\mu_{x} = \mu_{y}\) and we have

\[ \begin{aligned} T = \frac{(\overline{X} - \overline{Y})}{S_{p}\sqrt{\frac{1}{m} + \frac{1}{n}}} \sim t_{n+m-2} \end{aligned} \]

and we have the critical region defined as

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &T &> &t_{\alpha/2, n+m-2}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &T &\leq &t_{\alpha/2, n+m-2} \end{alignat} \]

and for the one sided hypothesis

\[ \begin{aligned} H_{0}: \mu_{x} \leq \mu_{y} \quad \text{versus} \quad H_{1}: \mu_{x} > \mu_{y} \end{aligned} \]

We reject \(H_{0}\) when the difference \(\mu_{x} - \mu_{y}\) is highly positive

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &T &> &t_{\alpha, n+m-2}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &T &\leq &z_{\alpha, n+m-2} \end{alignat} \]

Unknown and Unequal Variances¶

We consider the natural test statistic as follows

\[ \begin{aligned} \frac{(\overline{X} - \overline{Y}) - (\mu_{x} - \mu_{y})}{\sqrt{\frac{S_{x}^{2}}{n} + \frac{S_{y}^{2}}{m}}} \end{aligned} \]

Even when \(H_{0}\) is true, the above is not a simple distribution to solve for. If we make the additional assumption that \(n\) and \(m\) are large, then

\[ \begin{aligned} \frac{(\overline{X} - \overline{Y})}{\sqrt{\frac{S_{x}^{2}}{n} + \frac{S_{y}^{2}}{m}}} \sim \mathcal{N}(0, 1) \end{aligned} \]

and the same criteria for accepting and rejecting \(H_{0}\) discussed above are applicable, but after replacing population variances with sample variances.

Unknown and Unequal Variances¶

Suppose we want to observe the change in a quantity in a sample, after some kind of intervention. A simple example can be change in the mileage of a car after installation of a catalytic converter. Suppose we have \(n\) samples with us, and for each of the sample, we associate \(X_{i}\) with the measurement of the quantity before intervention and \(Y_{i}\) with the quantity post intervention. Note that \(X_{i}\) is not independent of \(Y_{i}\) because they come from the same \(i^{th}\) sample. Hence, the test discussed in section is not applicable.

Instead, we consider the quantity \(W = X - Y\) and assume that \(W_{i}\) come from a normal population. We can then consider the hypothesis

\[ \begin{aligned} H_{0}: \mu_{w} = 0 \quad \text{versus} \quad H_{1}: \mu_{w} \neq 0 \end{aligned} \]

Using the results derived in section, we have

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &\sqrt{n}\frac{\overline{W}}{S_{w}} &> &t_{\alpha/2, n-1}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &\sqrt{n}\frac{\overline{W}}{S_{w}} &\leq &t_{\alpha/2, n-1} \end{alignat} \]

One sided tests can be derived in exactly the same manner as section and the concepts discussed in section still hold true.