A 95 % confidence interval states that the population mean is greater than 152 and less than 160. If \(\sigma = 15\) and \(Z_{0.025} = 1.96\), then what sample size was used in the study?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 54
The margin of error E is half the width of the confidence interval:
\( \displaystyle E = \frac{160 - 152}{2} = 4 \)
For a 95% confidence interval, \(E = Z_{0.025} \times \frac{\sigma}{\sqrt{n}}\):
\( \displaystyle 4 = 1.96 \times \frac{15}{\sqrt{n}} \)
\( \displaystyle \sqrt{n} = \frac{1.96 \times 15}{4} = \frac{29.4}{4} = 7.35 \)
\(n = (7.35)^2 = 54.0225\)
The sample size is a whole number, and 54.0225 is nearest to 54. With n = 54, \(E = \frac{29.4}{\sqrt{54}} \approx 4.0008\), which gives the stated interval. So the sample size is 54.