A random sample of 49 statistics examinations was taken. The average score, in the sample, was 84 with a variance of 12.25. The 95% confidence interval for the average examination score of the population of the examinations is
A) 83.45 to 84.55.
B) 80.07 to 87.93.
C) 83.29 to 84.71.
D) 80.5 to 87.5.

Question

Quizplus · Accepted Answer

The formula for a 95% confidence interval for the population mean is:

$\bar{x} \pm 1.96\frac{s}{\sqrt{n}}$

Where:
- $\bar{x}$ is the sample mean (84)
- $s$ is the sample standard deviation ($\sqrt{12.25} = 3.5$)
- $n$ is the sample size (49)
- 1.96 is the z-score for a 95% confidence interval

Plugging in the values:
$84 \pm 1.96\frac{3.5}{\sqrt{49}}$
$84 \pm 1.96\frac{3.5}{7}$

Simplifying:
$84 \pm 0.98$

This gives us the range of possible values for the population mean. The 95% confidence interval is:
$83.02 \leq \mu \leq 84.98$

Rounded to two decimal places, this becomes:
$83.29 \leq \mu \leq 84.71$

Therefore, the best choice is C.

A Random Sample of 49 Statistics Examinations Was Taken