Question

The rejection region for a 5% level test of H₀ : μ ≥ 10 versus H₁ : μ < 10 is X¯ < 7.8. Find the rejection region for a 1% level test.

Answer 1

solution:

the given information as follows:

null and alternative hypothesis:

$H_0:\mu \geq 10$

$H_1:\mu < 10$

the test is a one tailed test

g= 0,05

rejecttion region = $\bar X<7.8$ (which mean if mean is less than 7.8 null will be rejected at 5%)

so critical value of z at for left tail from z table = -1.645

g= 0,05

so finding the standard error as follows:

$z = \frac{\bar X-\mu}{se}$

$Se=\frac{\bar X-\mu}{z}=\frac{7.8-10}{-1.645}=1.3374$

now we have to find the mean score for which null will be rejected at 1%

so

α = 0,01

critical value of z = $z_c=z_{0.01}=-2.33$

$z = \frac{\bar X-\mu}{se}$

$\bar X=(z * \sigma)+\mu=(-2.33*1.3374)+10=6.88$

so , rejection region at 1% = $\bar X < 6.88$