Question

A random sample n=30 is obtained from a population with unknown variance. The sample variance s² = 100 and the sample mean is computed to be 75. Test the hypothesis at α = 0.05 that the population mean equals 80 against the alternative that it is less than 80. The null hypothesis Ho: µ = 80 versus the alternative H1: Ho: µ < 80.

Calculate the test statistic from your sample mean. Then calculate the p-value for this test using the test stat and the information about H1. Because you do not know the population variance, you should use Table A.2 for the t distribution to estimate the p-value. The p-value for this hypothesis test is approximately ____________%. Answer in percent form. For example, 5% should be written as 5.0 and 0.5% should be written as 0.005.

Note: drawing a picture of the t distribution and the values on Table A.2 can help you figure out the p-value from your test statistic.(PLEASE SHOW ME THE STEP)

Answer 1

Ho :   µ =   80                  
Ha :   µ <   80       (Left tail test)          
                          
Level of Significance ,    α =    0.050                  
sample std dev ,    s =    10.0000                  
Sample Size ,   n =    30                  
Sample Mean,    x̅ =   75.0000                  
                          
degree of freedom=   DF=n-1=   29                  
                          
Standard Error , SE = s/√n =   10.0000   / √    30   =   1.8257      
t-test statistic= (x̅ - µ )/SE = (   75.000   -   80   ) /    1.8257   =   -2.74
                          
  
p-Value   =   0.0052   [Excel formula =t.dist(-2.74,29,true) ]             

p value = 0.52%
Decision:   p-value<α, Reject null hypothesis                       

Conclusion: There is enough evidence to conclude that true mean is less than 80