12.5: Two Means T Non-Pooled Test

Next, we will learn how to apply the Two Means \(T\) Non-Pooled Procedure to test a statistical claim about 2 population means when population standard deviations are unknown and cannot be assumed equal. Consider the following example.

A group of economists wants to the verify the claim that US workers have the average annual leave less than EU workers. Two samples of US and EU workers were obtained independently and analyzed. The sample of 32 US workers had the average annual leave of 20.4 days and the standard deviation 8.5 days. The sample of 38 EU workers had the average annual leave of 24.9 days and the standard deviation 4.3 days. Test the claim at \(1\%\) significance level.

Note that the sample means are \(\bar{x}_1=20.4\) days and \(\bar{x}_2=24.9\) days, the sample standard deviations are \(s_1=8.5\) days and \(s_2=4.3\) days, and the sample sizes are \(n_1=32\) and \(n_2=38\) for US and EU respectively.

Now, let’s identify the statistical claim that needs to be tested:

“US workers’ annual leave is less than EU workers’”

The claim is that the US workers’ annual leave is less than EU workers’, so we can symbolically express the claim as

\(\mu_1<\mu_2\) or \(\mu_1-\mu_2<0\)

Since the claim is about the 2 population means and the population standard deviations are unknown, we will use the 2 Means \(T\) Procedure but now we need to choose between pooled or non-pooled? We use the \(T\) Pooled Procedure when the standard deviations can be assumed equal, otherwise we use the \(T\) Non-Pooled Procedure.

To test the claim whether the standard deviations are the same we setup the 2 Variances \(F\) Test with \(\alpha=1\%\) (for convenience we chose the same alpha as in the problem but in general it doesn’t have to match and depends on the context).

\(H_0: \frac{\sigma^2_1}{\sigma^2_2}=1\)

\(H_a: \frac{\sigma^2_1}{\sigma^2_2}\neq1\)

\(f_0=\frac{s^2_1}{s^2_2}=\frac{8.5^2}{4.3^2}=3.908\)

\(\text{p-value}=0.001<\alpha=1\%\)

The conclusion is that we do have sufficient evidence to reject the null hypothesis, therefore we will believe that the variances are not the same and choose the \(T\) Non-Pooled Procedure.

Let’s check if all necessary assumptions are satisfied:

• The samples are simple random and independent
• The CLT must be applicable i.e. the populations are normal, or the samples are greater than 30
• The population standard deviations are unknown and not assumed equal

To use the \(T\) Non-Pooled Procedure, we must compute the degrees of freedom of the involved \(T\) distribution:

\(df=\frac{[\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}]^2}{\frac{(\frac{s^2_1}{n_1})^2}{n_1-1}+\frac{(\frac{s^2_2}{n_2})^2}{n_2-1}}=\frac{[\frac{8.5^2}{32}+\frac{4.3^2}{38}]^2}{\frac{(\frac{8.5^2}{32})^2}{32-1}+\frac{(\frac{4.3^2}{38})^2}{38-1}}=44.086\downarrow44\)

We will use the following template to perform the hypothesis testing:

In step 1, we will set up the hypothesis:

Since the claim is in the form of an inequality, therefore it must be set as an alternative hypothesis, therefore the null hypothesis is \(\mu_1-\mu_2=0\) and the test is left-tail, that is:

\(H_0: \mu_1-\mu_2=0\) \(H_a: \mu_1-\mu_2<0\)

LT

In step 2, we will identify the significance level:

The significance level can always be found in the text of the problem. In our case it is \(1\%\), thus:

\(\alpha=0.01\)

In step 3, we will find the test statistic using the formula:

\(t_0=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_1}{n_2}}}=\frac{20.4-24.9}{\sqrt{\frac{8.5^2}{32}+\frac{4.3^2}{38}}}=-2.716\)

In step 4, we will perform either the critical value approach or p-value approach to test the claim:

• In critical value approach, we construct the rejection region using the \(T\)-curve with \(df=44\):

RR: less than  \(-t_{0.01}=-2.414\)

• In p-value approach, we compute the p-value:

P-Value:  \(P(T<-2.716)=0.0047\)

In step 5, we will draw the conclusion:

• In the critical value approach, we must check whether the test statistic is in the rejection region or not. Our test statistic is \(-2.716\) and it is to the left of \(-2.414\), thus it is in the rejection region.
• In the p-value approach, we must check whether the p-value is less than the significance level or not. Our p-value is \(0.0047\) and it is less than \(\alpha=0.01\).

Both tests suggest that we DO reject the null hypothesis in favor of the alternative.

In step 6, we will interpret the results:

Under \(1\%\) significance level we DO have sufficient evidence to suggest that US workers have less annual leave than EU workers.

We discussed how to apply the Two Means \(T\) Non-Pooled Procedure to test a statistical claim about two population means with population standard deviations that are unknown and not assumed equal.