Question

Mean cholesterol level of the general population is known to be µ₀ = 175 with a known standard deviation σ = 40. Assume that n = 36 smokers were randomly selected and their cholesterol levels were recorded as x₁, . . . , x₂₅. It is speculated that mean cholesterol level of smokers (denoted by µ) may be different from µ₀ = 175.

1. 1. In testing H₀ : µ = 175 vs. H_A : µ /= 175 at the α = 0.05 level, you will reject H₀ either when x¯ < A or when x¯ > B. Calculate the values of A and B.
2. 2. The researcher is particularly interested in the possibility that the mean cholesterol level of smokers is µ₁ = 182. Thus, determination of the power of a z-test with this particular alternative (µ = µ₁ = 182) in mind is desirable. Which of the above two numbers (A and B) is the most relevant to calculating the power of the z test. Explain your choice both verbally and graphically.
   3. Compute the power of the above z test.

Answer 1

Given that

n=36

We have to test

i)

We reject H0 if

Since we have level of significance=0.0 5

So we allocate 0.025 left side and 0.025 right side

Hence

Now from Z table P(Z<-1.96)=0.025

Hence

Similarly

From Z table P(Z>1.96)=0.025

ii)

Since

Is more than 170 (Right to 170) hence right side of critical value so it's impossible that new mean will be in left side so B is most relevant to power.

iii)

We have to find the power

Power =P(rejecting H0|mean is 182)

Now