In the 2012 Boston Marathon, there were
- 7217 runners in their 40s with an average time of 255.2 minutes and a standard deviation of 43.7 minutes, and
- 4156 runners in their 50s with an average time of 270.8 minutes and a standard deviation of 44.7 minutes.
Use this information to test the null hypotesis that $\mu_{40}=\mu_{50}$ against the alternative hypthesis that
$\mu_{40} < \mu_{50}$, where $\mu_{40}$ denotes the average time of runners in their forties and $\mu_{50}$ denotes the average time of runners in their fifties. Be sure to
- Clearly state your hypothesis in terms of differences,
- compute the standard error,
- compute the test-statistic, and
- state the conclusion.
You might or might not feel the need to compute a $p$-value.