Errors in
Hypothesis Testing
2 TYPES OF ERRORS
TRUE CASE
HAis true HAis false
WE
Accept HA
SAY Do not Accept HA
CORRECT
TYPE I
ERROR
PROB = α
TYPE II
ERROR
CORRECT
PROB = β
α is set by the decision maker
β varies and depends on:
(1) α; (2) n; (3) the true value of 
Relationship Between  and 
•  is the Probability of making a Type II error
– i.e. the probability of not concluding HA is true when
it is
•  depends on the true value of 
• The closer the true value of  is to its
hypothesized value, the more likely we are of
not concluding that HA is true -i.e.  is large (closer to 1)
•  is calculated BEFORE a sample is taken
– We do not use the results of a sample to calculate 
CALCULATING 
• Example: If we take a sample of n = 49, with
 = 4.2, “What is the probability we will get a
sample from which we would not conclude  >
25 when  really = 25.5?” (Use  = .05)
REWRITE REJECTION REGION IN
TERMS OF x
The Hypothesis Test
Accept H A if z 
x  25
 1.645 or if
4.2
49
 4.2
x  25  1.645
 49

  25.987


CALCULATING  (cont’d)
• So when  = 25.5,
– If we get an x > 25.987, we will correctly conclude
that  > 25
– If we get an x < 25.987 we will not conclude that
 > 25 even though  really = 25.5
That’s a
TYPE II ERROR!!
P(Making this error) = 
CALCULATING  (cont’d)
• So what is P(not getting an x > 25.987 when 
really = 25.5?
That is P(getting an x < 25.987)?
Calculate z = (25.987 - 25.5)/(4.2/ 49)  .81
•  is the area to the left of .81 for a “>” test
• P(Z < .81) = .7910
β
“>” Test
Determining  When  = 25.5
DO NOT
ACCEPT HA
WRONG
ACCEPT HA
Prob = =.7910
RIGHT!
.7910
25.5
0
25.987
.81
X
Z
What is  When  = 27?
• So what is P(not getting an x > 25.987 when 
really = 27?
That is P(getting an x < 25.987)?
Calculate z = (25.987 - 27)/(4.2/ 49)  -1.69
•  is the area to the left of -1.69 for a “>” test
• P(Z < -1.69) = .0455
β
This shows that the further the true value of  is from the
hypothesized value of , the smaller the value of β; that is we
are less likely to NOT conclude that HA is true (and it is!)
“>” Test
Determining  When  = 27
DO NOT
ACCEPT HA
WRONG
ACCEPT HA
Prob = =.0455
RIGHT!
.0455
25.987
-1.69
27
0
X
Z
 for “<” Tests
• For n = 49,  = 4.2, “What is the
probability of not concluding that  < 27,
when  really is 25.5? (With  = .05)
The Hypothesis Test
Accept H A if z 
x  27
 1.645 or if
4.2
49
 4.2 
  26.013
x  27  1.645

 49 
• This time  is the area to the right of x
What is  When  = 25.5?
• So what is P(not getting an x < 26.013 when 
really = 25.5?
That is P(getting an x > 26.013)?
Calculate z = (26.013 – 25.5)/(4.2/ 49)  .86
•  is the area to the right of .86 for a “<” test
• P(Z > .86) = 1 - .8051 = .1949
β
“<” TEST
Determining  When  = 25.5
DO NOT
ACCEPT HA
ACCEPT HA
WRONG
RIGHT!
Prob = =.1949
.8051
.1949
25.5
0
26.013
.86
X
Z
 for “” Tests
• For n = 49,  = 4.2, “What is the probability of
not concluding that   26, when  really is 25.5?
(With  = .05)
The Hypothesis Test
x  26
Accept H A if z 
 1.96 or  1.96 or if
4.2
49
 4.2 
  24.824 or
x  26  1.96

 49 
 4.2 
  27.176
x  26  1.96

 49 
• This time  is the area in the middle between the
two critical values of x
What is  When  = 25.5?
• So what is P(not getting an x < 24.824 or
> 27.176 when  really = 25.5?
That is P(24.824 < x < 27.176)?
Calculate z’s = (24.824 – 25.5)/(4.2/ 49)  -1.13
and = (27.176 – 25.5)/(4.2/ 49 )  2.79
•  is the area in between -1.13 and 2.79 for a
“” test
β
• P(Z < 2.79) = .9974
• P(Z < -1.13) = .1292
P(-1.13 < Z < 2.79 = .9974 - .1292 = .8682
“” TEST
Determining  When  = 25.5
ACCEPT HA
DO NOT
ACCEPT HA
RIGHT!
WRONG
Prob = =.9974 –
.1292 =.8682
.8682
.9974
.1292
24.824
-1.13
25.5
0
27.176
2.79
X
Z
The Power of a Test = 1 - 
•  is the Probability of making a Type II error
– i.e. the probability of not concluding HA is true when
it is
•  depends on the true value of  and sample
size, n
• The Power of the test for a particular value of 
is defined to be the probability of concluding HA
is true when it is -- i.e. 1 - 
Power Curve Characteristics
• The power increases with:
– Sample Size, n
– The distance the true value of μ is from the
hypothesized value of μ
Power Curves For HA: μ  26
With n = 25 and n = 49
n = 49
n = 25
α = .05
Calculating  Using Excel
“> Tests”
Suppose H0 is  = 25;  = 4.2, n = 49,  = .05
“>” TESTS: HA:  > 25 and we want  when
the true value of  = 25.5
1) Calculate the critical x-bar value
= 25 + NORMSINV(.95)*(4.2/SQRT(49))
2) Calculate z = (critical x-bar -25.5)/ (4.2/SQRT(49))
3) Calculate the the probability of getting a zvalue < than this critical z value: -- this is 
=NORMSDIST(z)
Calculating  Using Excel
“< Tests”
Suppose H0 is  = 27;  = 4.2, n = 49,  = .05
“< TESTS”: HA:  < 27 and we want  when
the true value of  = 25.5
1) Calculate the critical x-bar value
= 27 - NORMSINV(.95)*(4.2/SQRT(49))
2) Calculate z = (critical x-bar -25.5)/ (4.2/SQRT(49))
3) Calculate the the probability of getting a zvalue > than the critical value: -- this is 
=1-NORMSDIST(z)
Calculating  Using Excel
“ Tests”
Suppose H0 is  = 26;  = 4.2, n = 49,  = .05
 TESTS: HA:   26 and we want  when the
true value of  = 25.5
1) Calculate the critical upper x-barU value and the
lower critical x-barL value
= 26 - NORMSINV(.975)*(4.2/SQRT(49))
= 26 + NORMSINV(.975)*(4.2/SQRT(49))
(x-barL)
(x-barU)
2) Calculate zU = (x-barU -25.5)/ (4.2/SQRT(49)) and
zL = (x-barL -25.5)/ (4.2/SQRT(49))
3) Calculate the the probability of getting an zvalue in between zL and zU - this is 
=NORMSDIST(zU) - NORMSDIST(zL)
β for “>” Tests
=B3+NORMSINV(1-B2)*(B5/SQRT(B6))
=(B8-B7)/(B5/SQRT(B6))
=NORMSDIST(B9)
=1-B10
β for “<” Tests
=B3-NORMSINV(1-B2)*(B5/SQRT(B6))
=(B8-B7)/(B5/SQRT(B6))
=1-NORMSDIST(B9)
=1-B10
β for “” Tests
=B3-NORMSINV(1-B2/2)*(B5/SQRT(B6))
=B3+NORMSINV(1-B2/2)*(B5/SQRT(B6))
=(B8-B7)/(B5/SQRT(B6))
=(B9-B7)/(B5/SQRT(B6))
=NORMSDIST(B11)-NORMSDIST(B10)
=1-B12
REVIEW
•
•
•
•
Type I and Type II Errors
 = Prob (Type I error)
 = Prob (Type II error) -- depends on , n and α
How to calculate  for:
– “>” Tests
– “<” Tests
– “” Tests
• Power of a Test at  = 1- 
• How to calculate  using EXCEL