Hypothesis Testing - Power of Test

Question

Suppose we are going to test null hypothesis whether the mean height of student grade 10

is 115cm against the alternative hypothesis as µ > 115 cm. Assuming that the height of

the student is normally distributed with mean µ and variance σ = 1. For a sample of 25 students,

(a) Find the critical region of the form of sample mean > k.

(b) Find β when µ = 116 cm.

You may use α = 0.05.

Answer 1

Hi Ashley,

a. Although the sample size is less than 25, since the problem tells us that the underlying distribution is normal, we can use the normal distribution to answer this question.

Since this is a right tailed test (Ha: Mu>115), the critical region will be in the right tail. If we look up the z value that represents the 95th percentile (where right tail equals alpha = 0.05, We see that the z value is 1.645

σ_xbar = 1/sqrt(25) = 0.2

Z=(k-µ)/σ_xbar

_{1.645 = (k-115)/0.2}

solving for k = 115.329

The critical region is k>115.329

b. β =P(fail to reject Ho | Ha true)

We need to calculate the area under the Ha curve that represents when you fail to reject Ho. That is the area under the curve defined as normal with µ=116, σ=0.2, that is less than 115.329.

We can calculate the z = (115.329 - 116)/0.2 = -3.355

P(Z< -3.355) = 0.0004

Please let me know if you would like a more detailed explanation