Question

A proposed null hypothesis states that there is no difference in the population mean heights of two neighboring districts. The difference of the sample means is 10 cm, and the standard deviation of the difference of sample means is 6 cm. Which conclusion can we draw at the 68% confidence level?

The population means of the two districts are different.

The population means of the two districts are not different.

The difference of the sample means of the two districts is 0.

The sample means of the two districts are not different.

Answer 1

We want the z-score to be between -1 and 1 because 68% of data that is normally distributed will be in this region:

http://www.wolframalpha.com/input/?i=z+score+calculator&a=FSelect_**NormalProbabilities-.dflt-&f2=-1&f=NormalProbabilities.z_-1&a=*FVarOpt.1-_***NormalProbabilities.z--.***NormalProbabilities.pr--.**NormalProbabilities.l-.*NormalProbabilities.r---.*--&a=*FVarOpt.2-_**-.***NormalProbabilities.mu--.**NormalProbabilities.sigma---.**NormalProbabilities.z---
The null hypothesis implies that the mean of the difference should be zero if there is no difference in the mean so at the 68% confidence interval, we would he expect data to be in this range:
$$
-1*\sigma\le0\le1*\sigma\\
-6\le0\le6\\
$$
But 10 cm is greater than 6, which is outside the 68% confidence interval.

What do you think this means?