9.03 Two sample t-tests
Two sample $t$t-tests
So far we have looked at testing hypotheses regarding whether a population mean appears to be correct, measured against a single sample using a single sample $t$t-test.
However sometimes we may wish to compare two samples. In this course we will conduct 'pooled' tests which means we assume the two samples come from populations with the same standard deviation (or variance). We must also assume that the populations are both normally distributed. The population standard deviation is estimated using technology from the known standard deviations of the two samples, $S_{x1}$Sx1 and $S_{x2}$Sx2. We compare the means of the two samples, which we call $\overline{x}_1$x1 and $\overline{x}_2$x2.
Our null hypothesis will be that there is no difference in the population means, $\mu_1=\mu_2$μ1=μ2. We can also write this as $\mu_1-\mu_2=0$μ1−μ2=0.
Our alternative hypothesis will either be single tailed, $\mu_1>\mu_2$μ1>μ2 or that $\mu_1<\mu_2$μ1<μ2.
Or it could be two tailed, where we say $\mu_1\ne\mu_2$μ1≠μ2, which means $\mu_1<\mu_2$μ1<μ2 or $\mu_1>\mu_2$μ1>μ2.
As for the single sample $t$t-test, a predetermined confidence interval is named and a graphing calculator is used to calculate the $p$p-values.
Graphics calculator instructions for two sample $t$t-test (given statistics):
| TI-nspire calculator instructions |
Casio fx-CG 50 calculator instructions |
TI-84 Plus CE calculator instructions |
|---|---|---|
| Choose A Calculate or Add Calculator | Press menu then select Statistics | Press stat |
| Press menu then select 6 Statistics | Press F3 for TEST then F2 for t | Select 4: 2-Samp TTest from the TESTS menu |
| Press 7 Stat Tests then 4 2-Sample t Test | Press F2 for 2-Sample | Use the Stats tab and type in the data. Ensure pooled is set to YES |
| Check the input method is Stats and type in the data. Ensure pooled is set to Yes | Type in the data. Ensure pooled is set to on. | Highlight Calculate |
| Press OK then OK to view the results | Scroll down to Execute and press EXE | Press enter to display the results |
Note that if you are given the raw data of the two samples then the calculator instructions are almost the same. You must first enter the data into List 1 and List 2 on your calculator. Then choose the option DATA as the input method, rather than STATS when using the calculator.
Worked examples
example 1
Buzz Electrics claim that the life spans of their candy bar vending machine light globes are normally distributed and that their new model bulbs last longer than their old model bulbs. A sample of $15$15 old model light globes are tested and found to have a mean of $1550$1550 hours and a sample standard deviation of $80$80 hours. A sample of $18$18 new model light globes are tested and found to have a mean of $1600$1600 hours and a sample standard deviation of $75$75 hours.
Conduct a two sample $t$t-test to determine if the new light globes last longer than the old light globes using a significance level of $0.05$0.05. Comment on your results.
Think: First define the hypothesis and variables. Be careful to be consistent with your use of $1$1 and $2$2. In this case, $1$1 = old globes and $2$2 = new globes.
Do: Let $\mu_1$μ1 be the population mean of the old globes. Let $\mu_2$μ2 be the population mean of the new globes.
The null hypothesis is $H_0$H0: $\mu_1=\mu_2$μ1=μ2 (the old globes are the same as the new globes).
The alternative hypothesis is $H_1$H1: $\mu_1<\mu_2$μ1<μ2 (the old globes last less than the new globes).
The level of significance is $\alpha=0.05$α=0.05 or $5%$5%.
Define $\overline{x}_1=1550$x1=1550 , $n_1$n1 = $15$15 and $S_{x1}=80$Sx1=80, $\overline{x}_2=1600$x2=1600 , $n_2$n2 = $18$18 and $S_{x2}=75$Sx2=75.
Enter the data into your calculator using the $2$2-sample $t$t-test application.
The calculator displays $t=-1.850231036$t=−1.850231036 and $p=0.0369160431$p=0.0369160431.
As $p<\alpha$p<α we say that there is enough evidence to reject $H_0$H0 on a $5%$5% level of significance. Therefore we can comment that the new globes do appear to last longer than the old globes.
example 2
A teacher is interested in whether there is a significant difference in the test scores of her female students in comparison to her male students. Conduct a $2$2-sample $t$t-test at a $10%$10% level of significance to investigate.
The students' test scores are given in the following table:
| Male | 95 | 78 | 68 | 95 | 98 | 79 | 98 | 86 | 78 | 89 | 89 | 94 |
| Female | 100 | 100 | 95 | 90 | 95 | 98 | 100 | 100 |
Think: First define the hypothesis and variables.
Do: Let $\mu_1$μ1 be the population mean of the male students. Let $\mu_2$μ2 be the population mean of the female students.
The null hypothesis is $H_0$H0: $\mu_1=\mu_2$μ1=μ2 (the results of the males and females are the same).
The alternative hypothesis is $H_1$H1: $\mu_1\ne\mu_2$μ1≠μ2 (the results are not the same).
The level of significance is $\alpha=0.1$α=0.1 or $10%$10%.
Do: Enter the male data into List 1 and the female data into List 2 of your calculator.
Then use the $2$2-sample $t$t-test application on your calculator and select DATA rather than STATS .
The calculator displays $t=-2.792775281$t=−2.792775281 and $p=0.0120213421$p=0.0120213421.
As $p<\alpha$p<α we reject the null hypothesis that the average scores are the same for both groups and we can say that there is a statistically significant difference between the test scores of the males compared to the females.
Reflect: Note that we cannot say that the female's scores are better or worse than the male's scores as that is not the hypothesis we are testing for. We can only comment that they are not equal. The meaning of the $10%$10% level of significance is that there is a $10%$10% chance that we incorrectly reject the null hypothesis.