We are comparing the means of two independent random samples, so essentially we are looking at the difference of the means=73.0-68.4=4.6.
We have two standard deviations to consider. To work out the standard error we need to combine the SD’s of the two samples taking into account the sample sizes. We use √(s₁²/n₁+s₂²/n₂), that is, the root mean square of the variances. Standard error=√(118.81/16+67.24/12)=√13.0290=3.6096 (approx).
We now have the test statistic, t=4.6/3.6096=1.2744.
We have small sample sizes (even when combined, the total is 28, which is less than 30) so we should use the formula below to find the degrees of freedom:
DOF=(s₁²/n₁+s₂²/n₂)²/[(s₁²/n₁)²/(n₁-1)+(s₂²/n₂)²/(n₂-1)].
DOF=13.0290²/[(118.81/16)²/15+(67.24/12)²/11]
DOF=25.995, which we can approximate to 26. It’s a one-tail test because we are considering 73.0>68.4 as the alternative hypothesis. So that’s the right tail of the distribution. For a 1-tail test at significance level 0.025, the critical value for DOF=26, the critical value is 2.056.
The test statistic is not as extreme as this critical value at 1.2744, so there is insufficient evidence to reject the null hypothesis at the given significance level. In other words, the difference between the means of the two samples is not big enough to suggest that the pulse rate for those people who do not exercise is greater than that of those people who do exercise, according to the samples examined.
