7.3.1.2  Unpooled Variances
7.3.1.2  Unpooled VariancesWhen the assumption of equal variances is not valid, we need to use separate, or unpooled, variances. The mathematics and theory are complicated for this case and we intentionally leave out the details.
We still have the following assumptions:
 The populations are independent
 Each population is either normal or the sample size is large.
If the assumptions are satisfied, then
\(t^*=\dfrac{\bar{x}_1\bar{x_2}0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}\)
will have a tdistribution with degrees of freedom
\(df=\dfrac{(n_11)(n_21)}{(n_21)C^2+(1C)^2(n_11)}\)
where \(C=\dfrac{\frac{s^2_1}{n_1}}{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}\).
 \((1\alpha)100\%\) Confidence Interval for \(\mu_1\mu_2\) for Unpooled Variances
 \(\bar{x}_1\bar{x}_2\pm t_{\alpha/2} \sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}\)

Where \(t_{\alpha/2}\) comes from the tdistribution using the degrees of freedom above.
Minitab^{®}
Minitab: Unpooled ttest
To perform a separate variance 2sample, tprocedure use the same commands as for the pooled procedure EXCEPT we do NOT check box for 'Use Equal Variances.'
 Choose Stat > Basic Statistics > 2sample t
 Select the Options box and enter the desired 'Confidence level,' 'Null hypothesis value' (again for our class this will be 0), and select the correct 'Alternative hypothesis' from the dropdown menu.
 Choose OK.
For some examples, one can use both the pooled tprocedure and the separate variances (nonpooled) tprocedure and obtain results that are close to each other. However, when the sample standard deviations are very different from each other, and the sample sizes are different, the separate variances 2sample tprocedure is more reliable.
Example 75: Grade Point Average
Independent random samples of 17 sophomores and 13 juniors attending a large university yield the following data on grade point averages (student_gpa.txt):
3.04  2.92  2.86  1.71  3.60 
3.49  3.30  2.28  3.11  2.88 
2.82  2.13  2.11  3.03  3.27 
2.60  3.13 
2.56  3.47  2.65  2.77  3.26 
3.00  2.70  3.20  3.39  3.00 
3.19  2.58  2.98 
At the 5% significance level, do the data provide sufficient evidence to conclude that the mean GPAs of sophomores and juniors at the university differ?
There is no indication that there is a violation of the normal assumption for both samples. As before, we should proceed with caution.
Now, we need to determine whether to use the pooled ttest or the nonpooled (separate variances) ttest. The summary statistics are:
Variable 
Sample size 
Mean 
Standard Deviation 

sophomore 
17 
2.840 
0.52 
junior 
13 
2.981 
0.3093 
The standard deviations are 0.520 and 0.3093 respectively; both the sample sizes are small, and the standard deviations are quite different from each other. We, therefore, decide to use an unpooled ttest.
The null and alternative hypotheses are:
\(H_0\colon \mu_1\mu_2=0\) vs \(H_a\colon \mu_1\mu_2\ne0\)
The significance level is 5%. Perform the 2sample ttest in Minitab with the appropriate alternative hypothesis.
Remember, the default for the 2sample ttest in Minitab is the nonpooled one. Minitab generates the following output.
Two sample T for sophomores vs juniors
N  Mean  StDev  SE Mean  

sophomore  17  2.840  0.52  0.13 
junior  13  2.981  0.309  0.086 
95% CI for mu sophomore  mu juniors: (0.45, 0.173)
TTest mu sophomore = mu juniors (Vs no =): T = 0.92
P = 0.36 DF = 26
Since the pvalue of 0.36 is larger than \(\alpha=0.05\), we fail to reject the null hypothesis.
At 5% level of significance, the data does not provide sufficient evidence that the mean GPAs of sophomores and juniors at the university are different.
95% CI for mu sophomore mu juniors is;
(0.45, 0.173)
We are 95% confident that the difference between the mean GPA of sophomores and juniors is between 0.45 and 0.173.