Question

A one-sided test of the null hypothesis μ = 24 versus the alternative  μ = 34 has power equal to 0.73. Will the power for the alternative  μ = 39be higher or lower than 0.73?

Draw a picture and use this to explain your answer (choose one of the following):

(1) A picture shows us that the area under the curve where H₀ is rejected is larger for the normal distribution with mean equal to the alternative μ = 39 than for the normal distribution with mean equal to the alternative μ = 34. This indicates that larger differences have smaller power and therefore are harder to detect.

(2) A picture shows us that the area under the curve where H₀ is rejected is larger for the normal distribution with mean equal to the alternative μ = 39 than for the normal distribution with mean equal to the alternative μ = 34. This indicates that larger differences have greater power and are easier to detect.    

(3) A picture shows us that the area under the curve where H₀ is not rejected is larger for the normal distribution with mean equal to the alternative μ = 39 than for the normal distribution with mean equal to the alternative μ = 34. This indicates that larger differences have greater power and are easier to detect.

(4) A picture shows us that the area under the curve where H₀ is not rejected is larger for the normal distribution with mean equal to the alternative μ = 34 than for the normal distribution with mean equal to the alternative μ = 39. This indicates that smaller differences have greater power and are easier to detect.

(5) A picture shows us that the area under the curve where H₀ is rejected is larger for the normal distribution with mean equal to the alternative μ = 34 than for the normal distribution with mean equal to the alternative μ = 39. This indicates that smaller differences have smaller power and therefore are harder to detect.

Answer 1

If at the power is 0.73

that means the critical value is also at 34. Now if we change the alternative mean to 39, the critical value stays the same at 34.

However, the graph itself shifts so that the new center is 39. You can see that the power increases. This makes sense since the further away the null and alternative mean are, if the alternative mean is correct, a random sample should produce a sample average close to µa and not µo. The likely hood of rejecting the null increases.

Option 2 is right