Question

If a$95\%$confidence interval for the population mean was reported to be 160 to 170 and$σ = 25$, then the size of the sample used in this study is:

• A. 96
• B. 125
• C. 54
• D. 81

Answer 1

Answer: (A)

96

Answer 2

The formula for the margin of error in a confidence interval is:
$E = Z \cdot \frac{\sigma}{\sqrt{n}},$
where $E = \text{Width of the interval} \div 2$, $Z = 1.96$, and $\sigma = 25$.
The width of the confidence interval is:
$170 - 160 = 10 \quad \Rightarrow \quad E = \frac{10}{2} = 5.$
Substitute into the formula:
$5 = 1.96 \cdot \frac{25}{\sqrt{n}}.$
Solve for $n$:
$\sqrt{n} = \frac{1.96 \cdot 25}{5} = 9.8 \quad \Rightarrow \quad n = 9.8^2 = 96.04.$
Thus, the sample size is $n = 96$.