A 95% confidence interval for a population mean was reported to be 34 to 38. If variance, \( \sigma^2=100 \), then the appropriate sample size used in this study is : (given \( Z_{0.025}=1.96 \))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 96
The 95% confidence interval for the mean is \( \bar{x}\pm Z_{0.025}\cdot\frac{\sigma}{\sqrt{n}} \). The interval 34 to 38 has centre 36 and half-width 2.
\( \displaystyle \sigma=\sqrt{100}=10 \)
\( \displaystyle 1.96\times\frac{10}{\sqrt{n}}=2 \)
\( \displaystyle \sqrt{n}=\frac{19.6}{2}=9.8 \)
\( \displaystyle n=(9.8)^2=96.04\approx 96 \)
Tip: The margin of error is half the length of the interval, not its length; using 4 instead of 2 here would give a sample size four times too small.