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 with a Standard Deviation of 15000 (USA Today, August 24, 1995). A researcher for the state of South Carolina wants to test the following hypothesis:
where μ is the mean annual income for a resident of South Carolina.
The researcher gathers information from a sample of 625 residents of South Carolina and finds a sample mean of 17,076 ( =17,076) with a sample standard deviation (s) equal to 15,500.
a. What is the appropriate conclusion pertaining to the hypothesis formulated above? Use a .01 level of significance.
b. Construct a 99% Confidence Interval for the value of the mean pertaining to the South Carolina residents' income.
c. How would your answer to part a. if you did not know the value of the population standard deviation and your sample size was only 25?.
In all these problems, show the area under the curve to graphically demonstrate your answers.