A 95% confidence interval for a population mean was reported to be 152 to 160. If standard deviation \(\sigma = 15\), then the sample size is: (\(Z_{.025}=1.96\))
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 54
The confidence interval is \(\bar{x} \pm E\), where the margin of error is \(E = Z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). The sample mean is the centre of the interval and E is its half-width:
\( \displaystyle \bar{x} = \frac{152 + 160}{2} = 156 \)
\(E = 160 - 156 = 4\)
So
\( \displaystyle 1.96 \times \frac{15}{\sqrt{n}} = 4 \)
\( \displaystyle \sqrt{n} = \frac{1.96 \times 15}{4} = 7.35 \)
\(n = (7.35)^2 = 54.02\)
A sample size must be a whole number. Here n is taken as the nearest whole number, 54: the interval was reported from a real sample, and with \(n = 54\) the half-width is \(\frac{29.4}{\sqrt{54}} = 4.001\), practically 4. (Rounding up to 55 is used when planning a new sample, so that E does not exceed 4; 55 is not an option here.)
So the sample size is 54.