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 years mean chess 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 use a Z-test with 10% level of significance and a sample size of 65.

1.) Find the power if in fact the last years mean cheese consumption is 36.3 lb

2.) Find the power if in fact the last years mean cheese consumption is 33.7 lb

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

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  
Probability of type 2 error ( ß ) = P ( X̅ < 34.1 | µ = 36.3 ) = 0.0051
Power of test is ( 1 - ß ) = 0.9949

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
Probability of type 2 error ( ß ) = P ( X̅1 < 34.1 | µ = 33.7 ) = 0.6799
Power of test is = ( 1 - ß ) = 0.3201

Part 3)

As the value of mean decreases, power of the test also decreases.

So the relationship of mean is direclty proportional to power.