Question

The U.S. Department of Agriculture reports in Food Consumption, Prices, and Expenditures that the average American consumed 33 lb of cheese in 2010. Suppose that you want to decide whether last year’s mean cheese consumption is greater than the 2010 mean. The preliminary research indicated that population standard deviation is known to be 6.9 lb. Suppose that you decide to use a z-test with 10% level of significance and a sample size of 65.

(1) Find the power if in fact the last year’s mean cheese consumption is 36.2 lb.

(2) Find the power if in fact the last year’s mean cheese consumption is 33.7 lb.

(3) Use results from (1) & (2) above to interpret the relationship between true mean and power.

(please do not use Minitab to show results, I would like step by step)

Answer 1

Part 1)

The values of sample mean X̅ for which null hypothesis is rejected
Z = ( X̅ - µ ) / ( σ / √(n))
Critical value Z(α/2) = Z( 0.1 /2 ) = ± 1.282
1.645 = ( X̅ - 33 ) / ( 6.9 / √( 65 ))
X̅ >= 34.1
P ( X̅1 < | µ = 36.2 ) =
Probability of type 2 error ( ß ) = 0.0071
Power of test is = ( 1 - ß ) = 0.9929

Part 2)

The values of sample mean X̅ for which null hypothesis is rejected
Z = ( X̅ - µ ) / ( σ / √(n))
Critical value Z(α/2) = Z( 0.1 /2 ) = ± 1.282
1.645 = ( X̅ - 33 ) / ( 6.9 / √( 65 ))
X̅ >= 34.1
P ( X̅1 < | µ = 33.7 ) =
Probability of type 2 error ( ß ) = 0.6799
Power of test is = ( 1 - ß ) = 0.3201

Part 3)

As mean cheese consumption decreases, power of the test also decreases.

We can comment on the relationship as true mean is directly proportional to power.

any doubts please ask ! thank you :)