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)

Reject $H_0$ if $∣Z∣>1.96$. Since $Z=−2.00$, reject $H_0$.

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 $H_0$.