Question

The Bureau of Economic Analysis in the U.S. Department of Commerce reported that the mean annual income for a resident of North Carolina is $18,688 (USA Today, August 24, 1995). A researcher for the state of South Carolina wants to see if the mean annual income for a resident of South Carolina is different. A sample of 400 residents of South Carolina shows a sample mean annual income of $16,860 and the population standard deviation is assumed to known,  =$14,624. Use a 0.05 level of significance, the researcher wants to test the following hypothesis.

H0:  = 18,688 Ha:   18,688

a) Use confidence interval approach to test the hypothesis?

b)What is your conclusion?

C)What are three rejection rules (You have used confidence interval approach in Question 2)?

D)Do three rejection rules lead to the same conclusion? What is your conclusion?

Answer 1

Let X be the annual income      
Let n be the sample size      
       
Null and alternative hypotheses are      
Ho : μ = 18688      
Ha : μ ≠ 18688       
       
X̅ = 16860                      Sample Mean    
n = 400                      Sample Size    
σ = 14624                  Population Standard Deviation    
α = 0.05,     α/2 = 0.025      
a) Confidence interval for mean is given by      

For confidence interval, given α = 0.05, α/2 = 0.025      
From the z-tables, or Excel function NORM.S.INV(α/2)      
z = NORM.S.INV(0.025) = 1.96    (We take the positive value for calculations)  
Confidence interval is      

= (15426.85, 18293.15)      
Confidence interval is ($15426.85, $18293.15)      
Since $18688 is not within the confidence interval ($15426.85, $18293.15)      
we reject the null hypothesis      
       
b) Conclusion:      
There is sufficient statistical evidence at 5% level of significance to conclude that       
the mean annual income for a resident of South Carolina is different from $18688      
       
c) The 3 rejection rules are      
1) Reject Ho if the sample mean is less than $15426.85      
2) Reject Ho if the sample mean is greater than $18293.15      
3) Reject Ho if the sample mean is not within ($15426.85, $18293.15)      
       
d) Since $18688 is not within the confidence interval ($15426.85, $18293.15)      
we reject the null hypothesis (Rules 2 and 3 are applicable here)      
Conclusion:      
There is sufficient statistical evidence at 5% level of significance to conclude that       
the mean annual income for a resident of South Carolina is different from $18688