Transcript Ch8.2 Population Mean Test Case I: A Normal Population With Known Null hypothesis:   0 x  0 z  / n H0 : Test statistic value: Alternative.

Ch8.2 Population Mean Test
Case I: A Normal Population With Known
Null hypothesis:
  0
x  0
z
 / n
H0 :
Test statistic value:
Alternative Hypothesis Rejection Region for Level  Test
Ha :
  0
Ha :
  0
Ha :
  0
z  z
z   z
z  z / 2 or z   z / 2
Ch8.2
Recommended Steps in Hypothesis-Testing Analysis
1. Identify the parameter of interest and describe it in the
context of the problem situation.
2. Determine the null value and state the null hypothesis.
3. State the alternative hypothesis.
4. Give the formula for the computed value of the test
statistic.
5. State the rejection region for the selected significance
level
6. Compute any necessary sample quantities, substitute
into the formula for the test statistic value, and
compute that value.
7.
Decide whether H0 should be rejected and state this
conclusion in the problem context.
Ch8.2
Type II Probability  (  )for a Level  Test
Type II Probability  (  )
0    

Ha :   0
  z 


/
n


0    

1     z 
Ha :   0

/ n 

0    
0    


  z / 2 
Ha :   0
     z / 2 


/
n

/
n




The sample size n for which a level  test also has  (  )  
at the alternative value   is
Alt. Hypothesis
   ( z  z  )  2
 
0    
 

n  
2
   ( z / 2  z  ) 






0


Ch8.2
one-tailed test
two-tailed test
Case II: Large-Sample Tests
When the sample size is large, the z tests for case I are
modified to yield valid test procedures without requiring
either a normal population distribution or a known  .
Large Sample Tests (n > 40)
Test Statistic:
Z 
X  0
S/
n
The use of rejection regions for case I results in a test
procedure for which the significance level is
approximately .
Ch8.2
Case III: A Normal Population Distribution
If X1,…,Xn is a random sample from a normal distribution,
the standardized variable
X 
T 
S/
n
has a t distribution with n – 1 degrees of freedom.
H0 :   0
Test statistic value: t  x   0
s/ n
Null hypothesis:
Alternative Hypothesis Rejection Region for Level
Ha :   0
t  t ,n1
Ha :   0
t  t ,n1
Ha :   0
 Test
t  t / 2,n1 or t  t / 2,n1
Ch8.2