Question

select the correct answer.
a proposed null hypothesis states that there is no difference in the population mean heights of males of two neighboring towns. the sample mean difference is found to be 10 cm, and the standard deviation of the difference of the sample means is 6 cm. which statement is true?
a. the null hypothesis must be rejected if we choose the 68% confidence level.
b. the null hypothesis must be rejected if we choose the 95% confidence level.
c. the null hypothesis must be rejected if we choose the 99.7% confidence level.
d. the null hypothesis must be rejected if we choose the 100% confidence level.

Explanation:

Step1: Recall Confidence Intervals

For a normal distribution, the confidence intervals are: 68% within \( \mu \pm \sigma \), 95% within \( \mu \pm 2\sigma \), 99.7% within \( \mu \pm 3\sigma \), and 100% would include all values (but practically, 100% confidence interval is the entire range of possible values, which is not useful here). The null hypothesis is \( H_0: \mu_1 - \mu_2 = 0 \). The sample mean difference is \( \bar{x}_1 - \bar{x}_2 = 10 \) cm, and the standard deviation of the difference of sample means (standard error) is \( \sigma_{\bar{x}_1 - \bar{x}_2} = 6 \) cm.

Step2: Analyze Each Option

• Option A (68% confidence level): The interval is \( 0 \pm 1\times6 = (-6, 6) \). The sample mean difference (10) is outside this interval. So, we reject \( H_0 \) at 68% confidence.
• Option B (95% confidence level): The interval is \( 0 \pm 2\times6 = (-12, 12) \). The sample mean difference (10) is inside this interval. So, we do not reject \( H_0 \) at 95% confidence.
• Option C (99.7% confidence level): The interval is \( 0 \pm 3\times6 = (-18, 18) \). The sample mean difference (10) is inside this interval. So, we do not reject \( H_0 \) at 99.7% confidence.
• Option D (100% confidence level): A 100% confidence interval would include all real numbers (since we can't be 100% sure otherwise), so the sample mean difference (10) would be inside, and we do not reject \( H_0 \).

Answer:

A. The null hypothesis must be rejected if we choose the 68% confidence level.