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?

54

The correct answer is Option (4) → 54

$\text{Given:}$

$\sigma = 15,\quad z_{0.025} = 1.96$

$\text{Confidence interval: }(152,\ 160)$

$\text{The margin of error is:}$

$E = \frac{\text{Upper limit} - \text{Lower limit}}{2} = \frac{160 - 152}{2} = \frac{8}{2} = 4$

$\text{Formula for margin of error in a confidence interval:}$

$E = z_{0.025} \cdot \frac{\sigma}{\sqrt{n}}$

$\text{Substitute the values:}$

$4 = 1.96 \cdot \frac{15}{\sqrt{n}}$

$\text{Multiply both sides by }\sqrt{n}:$

$4\sqrt{n} = 1.96 \cdot 15$

$4\sqrt{n} = 29.4$

$\text{Divide both sides by 4:}$

$\sqrt{n} = \frac{29.4}{4} = 7.35$

$\text{Square both sides:}$

$n = (7.35)^2 = 54.0225$

$\text{Since sample size must be a whole number, round up:}$

$n = 54$