Question

A union of restaurant and foodservice workers would like to estimate this year's mean hourly wage for foodservice workers in the U.S. Last year's mean hourly wage was 58.14, and there is reason to believe that this year's value is greater than last year's The union decides to do a statistical test to see if it can be concluded that the mean has increased. The union chooses a random sample of 125 wages from this year Suppose that the population of hourly wages of foodservice workers in the U.S. has a standard deviation of 1.15 and that the union performs its hypothesis test using the 0.01 level of significance Based on this information, answer the questions below. Carry your intermediate computations to at least four decimal places, and round your responses as indicated (if necessary, consulta list of formulas.) od His What are the land alternative hypotheses that the wnion should use for the test? What is the probability that the union commits a Type error? Round your response to that we decimal 0 probably that the ones theollypothesis? and your tast two decimal 0

Answer 1

Claim : The mean hourly wage of U.S is greater than 8.14

он
: µ less than equal to 8.14 vs
H
: µ greater than 8.14  

Given : σ = 1.15 , n = 125 ,So σ/√n = 0.1029  

Probability that the union commits a type I error α = 0.01

As H₁ contain > sign , this is right tail test, so critical value would be positive.

So 1 - α = 0.99 therefore critical value Z= 2.33

= ( Z*σ/√n ) + µ = ( 2.33*0.1029) + 8.14

T

= 8.3798

T

Rejection region : Reject H0 , if
T
≥ 8.3798 OR accept H0 , if
T
< 8.3798

Now we are asked to find the probability of type II error for µ =8.43

P( Accept the null hypothesis , when µ = 8.43 )

= P( < 8.3798 , when µ = 8.43 )  

T

=

T-ME P 8.3798 – 8.43 0.1029 01

= P( z < -0.49 )

Probability that union accept null hypotheis = 0.3121

If we increase the level of significance , the type II error decreases as a result power of the test increases.

Therefore, The power of the second test is greater than the power of the original test.