Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).
Assertion (A): If a simple random sample of 100 items for a population standard deviation $σ = 8$ resulted in sample mean $\bar x= 45$, then 95% confidence interval for population mean is (43.43, 46.57). (use $Z_{0.025} = 1.96$)
Reason (R): Interval estimation of population mean, $μ=\bar x ± Z_{α/2}.\frac{σ}{\sqrt{n}}$
Select the correct answer from the given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
Given $n = 100, σ = 8, \bar x = 45$.
Confidence level = 95%
$⇒ 1-α= 0.95⇒ α = 0.05⇒\frac{α}{2}=0.025$
$∴Z_{α/2}=Z_{0.025} = 1.96$
Now, margin of error $=Z_{α/2}.\frac{σ}{\sqrt{n}}= 1.96 ×\frac{8}{\sqrt{100}}= 1.96 × 0.8 = 1.57$
$∵ μ=\bar x$ ±margin of error $= 45 ± 1.57$.
So, confidence interval is (45 -1.57, 45+ 1.57) i.e. (43.43, 46.57).
∴ Assertion is true.
Also, Reason is true and Reason is the correct explanation of Assertion.