Consider the following hypothesis test:
$H_0: μ = 15$
$H_a: μ ≠ 15$.
A sample of 50 provided a sample mean of 14.15. The population standard deviation is 3. What is the rejection rule using the critical value? What is your conclusion? ($α$ = 0.05)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Reject $H_0$ if $∣Z∣>1.96$. Since $Z=−2.00$, reject $H_0$.
Given $μ_0 = 15, n = 50, \bar x = 14.15, σ = 3$ and $α=0.05$
$Z=\frac{\bar x-μ_0}{\frac{σ}{\sqrt{n}}}=\frac{14.15-15}{\frac{3}{\sqrt{50}}}=\frac{-0.85×\sqrt{50}}{3}$
$=-2.003$
$∴Z=-2$
Reject $H_0$ if $Z≤-Z_{α/2}$
$∵-Z_{α/2}=-Z_{0.025}=-1.96$
$∵-2<-1.96$
So, reject $H_0$.