A 95% confidence interval for a population mean was reported to be 152 to 160. If $σ = 15$, what sample size was used in this study?

54

The correct answer is Option (3) → 54

Given $σ = 15$, confidence level = 95%, confidence interval (152, 160).

Let the sample mean be $\bar x$ and margin of error be E,

then $\bar x - E=152$  ...(i)

and $\bar x + E = 160$  ...(ii)

Subtracting equation (i) from (ii), we get

$2E=8⇒E=4$

Also, $1-α = 0.95⇒α = 0.05⇒\frac{α}{2}=0.025$

$⇒ Z_{α/2}=Z_{0.025}⇒ Z_{α/2} = 1.96$

We know that margin of error = $Z_{α/2}.\frac{σ}{\sqrt{n}}$

$⇒4 = 1.96 ×\frac{15}{\sqrt{n}}$

$⇒\sqrt{n}=\frac{1.96 × 15}{4}$

$⇒\sqrt{n} = 0.49 × 15$

$⇒\sqrt{n} = 7.35⇒n= (7.35)^2$

$⇒ n = 54.0225⇒n=54$

So, sample size is 54.