Question

select the correct answer.
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?
a. the population means of the two districts are different.
b. the population means of the two districts are not different.
c. the difference of the sample means of the two districts is 0.
d. the sample means of the two districts are not different.

Explanation:

Step1: Recall the 68 - 95 - 99.7 rule

For a normal distribution, about 68% of the data lies within \( \mu\pm\sigma \) (where \( \mu \) is the mean and \( \sigma \) is the standard deviation). Here, we are dealing with the difference of sample means. The null hypothesis assumes the population mean difference is 0 (no difference in population means). The difference of sample means is 10 cm, and the standard deviation of the difference of sample means is 6 cm.

Step2: Check the range for 68% confidence

The range for 68% confidence around the null hypothesis (population mean difference = 0) would be \( 0\pm6 \), i.e., from - 6 to 6. The sample mean difference is 10, which is outside this range. So, we reject the null hypothesis. Rejecting the null hypothesis (which said there is no difference in population means) means we conclude that the population means of the two districts are different.

Answer:

A. The population means of the two districts are different.