Research question: Do the mean commute times in Atlanta and St. Louis differ in the population?
StatKey was used to construct a randomization distribution:
Step 1: Check assumptions and write hypotheses
From the given StatKey output, the randomization distribution is approximately normal.
\(H_0: \mu_1-\mu_2=0\)
\(H_a: \mu_1 - \mu_2 \ne 0\)
Step 2: Compute the test statistic
\(test\;statistic=\dfrac{sample\;statistic - null \; parameter}{standard \;error}\)
The observed sample statistic is \(\overline x _1 - \overline x _2 = 7.14\). The null parameter is 0. And, the standard error, from the StatKey output, is 1.136.
\(test\;statistic=\dfrac{7.14-0}{1.136}=6.285\)
Step 3: Determine the p value
The p value will be the area on the z distribution that is more extreme than the test statistic of 6.285, in the direction of the alternative hypothesis:
This was a two-tailed test. The area in the two tailed combined is 0.000000. Theoretically, the p value cannot be 0 because there is always some chance that a Type I error was committed. This p value would be written as p < 0.001.
Step 4: Make a decision
The p value is smaller than the standard 0.05 alpha level, therefore we reject the null hypothesis.
Step 5: State a "real world" conclusion
There is evidence that the mean commute times in Atlanta and St. Louis are different in the population.