Question

X is normally distributed with the following data (25.8, 36.6, 26.3, 21.8, 27.2). Select the correct statement about X ($t_{\text{crit},\alpha=0.05,4}=2.132$):

• A. Population mean $\le 25$ with 95% confidence
• B. Population mean $\le 25$ with 100% confidence
• C. Population mean $> 25$ with 95% confidence
• D. Population mean $> 25$ with 100% confidence

Hint:

A 95% confidence interval tells you the range in which the true mean may lie; if 25 is within the interval, it cannot be rejected at 95% confidence.

Answer 1

The Correct Option is A

Sample values: 25.8, 36.6, 26.3, 21.8, 27.2.
Compute sample mean:
\[ \bar{x} = 27.54 \] Compute sample standard deviation $s \approx 5.68$.
The 95% confidence interval for the population mean:
\[ \bar{x} \pm t_{\text{crit}} \frac{s}{\sqrt{5}} = 27.54 \pm 2.132 \times 2.54 = 27.54 \pm 5.41 \] \[ \Rightarrow (22.13,32.95) \] Since the entire CI lies above 22.13 and includes 25, it is correct to say:
Population mean $\le 25$ is not rejected at 95% confidence.
Thus option (A) is correct.
Final Answer: Population mean $\le 25$ with 95% confidence