1. Statistics for Managers Using Microsoft® Excel 7th Edition
Chapter 9
Fundamentals of Hypothesis Testing:
One Sample Tests
Statistics for Managers Using Microsoft Excel, 7e © 2014 Pearson-Prentice-Hall, Inc Philip A. Vaccaro , PhD

2. Learning Objectives

3. The Hypothesis population mean ( μ ) pop. proportion ( π )
A hypothesis is a claim (assumption) about a population parameter:
population mean ( μ )
pop. proportion ( π )

4. The Null Hypothesis, H0
it is written in terms of the population

5. The Null Hypothesis, H0

6. The Alternative Hypothesis, H1

7. The Alternative Hypothesis, H1

8. The Hypothesis Testing Process
Claim: The population mean age is 50.
H0: μ = 50, H1: μ ≠ 50
Sample the population and find sample mean.
Population
Sample

9. The Hypothesis Testing Process_

10. The Hypothesis Testing Process
Sampling Distribution of X X 20
μ = 50
If H0 is true

11. The Test Statistic and Critical Values

12. The Test Statistic and Critical Values
Distribution of the test statistic
Region of Rejection
Region of Rejection
Do Not Reject Ho
Critical Values
The
critical
values
are stated
in either
‘z’ or ‘t’

13. Errors in Decision Making
Type I Error
Reject a true null hypothesis
Considered a serious type of error
The probability of a Type I Error is ‘’
Called level of significance of the test
Set by researcher in advance:
Type II Error
Failure to reject false null hypothesis
The probability of a Type II Error is ‘β’
within your control or dictated by management

14. Errors in Decision Making

15. Level of Significance, α
The probability of rejecting Ho when it is true
Claim: The population mean age is 50.
a/2
a/2
H0: μ = H1: μ ≠ 50
Two-tail test
H0: μ ≤ 50 H1: μ > 50 a
Upper-tail test
typical
values
are
α =.01
α =.02
α =.05
α =.10
H0: μ ≥ 50 H1: μ < 50 a
Lower-tail test

16. Hypothesis Testing: σ Known_

17. Hypothesis Testing: σ Known
H0: μ = 3 H1: μ ≠ 3
/2
/2 X 3
Reject H0
Do not reject H0
Reject H0 Z -Z +Z
Lower critical value
Upper critical value

18. Hypothesis Testing: σ Known

19. Hypothesis Testing: σ Known_

20. Hypothesis Testing: σ Known
Is the test statistic in the rejection region?
= .05/2
 = .05/2
.025
.025
a = .05
level of
significance
Reject H0
Do not reject H0
Reject H0
-Z= -1.96
+Z= +1.96 Z
0.06
- 1.9
.0250
Here, Z = -2.0 < -1.96, so the test statistic is in the rejection region Z
0.06
+ 1.9
.9750

21. Hypothesis Testing: σ Known

22. Hypothesis Testing: σ Known

23. Hypothesis Testing: σ Known

24. Hypothesis Testing: σ Known , the ‘p’- Value Approach
* Because we can reject or not reject Ho “at a glance“ .

25. Hypothesis Testing: σ Known p-Value Approach

26. Hypothesis Testing: σ Known p-Value ApproachZ
0.00
- 2.00
.0228 Z
0.00
+ 2.00
.9772
.0228
/2 = .025
-1.96
-2.0 Z
1.96
2.0
Z = .8
√100
= =
.08
(.9772)
Z = .8
√100
= =
.08
Ho: μ = 3.00

27. Hypothesis Testing: σ Known p-Value Approach
probability
of seeing a
sample mean
of 2.84 or less,
or 3.16 or
more , if
the population
mean is
really 3.0
is only 4.56%
Compare the p-value with 
If p-value <  , reject H0
If p-value   , do not reject H0
.0228
/2 = .025
-1.96
-2.0 Z
1.96
2.0

28. Hypothesis Testing: σ Known Confidence Interval Connections
Simpler,
faster
hypothesis
test If
a = .05,
construct
a 95%
confidence
interval
Can
only be
used with
a two-tail
hypothesis
test !

29. Hypothesis Testing: σ Known One Tail Tests

30. Hypothesis Testing: σ Known Lower Tail Tests
There is only one critical value, since the rejection area is in only one tail. α
Reject H0
Do not reject H0 Z -Z μ X
Critical value

31. Hypothesis Testing: σ Known Upper Tail Tests
There is only one critical value, since the rejection area is in only one tail. α Z
Do not reject H0
Reject H0 Z X μ
Critical value

32. Hypothesis Testing: σ Known Upper Tail Test Example
Form the hypothesis test:

33. Hypothesis Testing: σ Known Upper Tail Test Example
Suppose that  = .10 is chosen for this test
Find the rejection region:
Reject H0
Do not reject H0
 = .10 Z
1- = .90
The
entire
a = .10
goes
to the
upper
tail Z
0.08
+ 1.2
.8997

34. Hypothesis Testing: σ Known Upper Tail Test Example
What is Z given a = 0.10?
.90
.10
.08 Z
.07
.09
a = .10
1.1
.8790
.8810
.8830
.90
1.2
.8980
.8997
.9015 z
1.28
1.3
.9147
.9162
.9177
Critical Value
= 1.28

35. Hypothesis Testing: σ Known Upper Tail Test Example-

36. Hypothesis Testing: σ Known Upper Tail Test Example
Reach a decision and interpret the result:
Reject H0
1- = .90
 = .10
Do Not Reject Ho
1.28
Z = .88

37. Hypothesis Testing: σ Known Upper Tail Test Example
Calculate the p-value and compare to ‘’
p-value = .1894 Z
0.08
+ .8
.8106
Reject H0
 = .10
Z = = 1.1
10 / √
= .88
= .1894
Do not reject H0
Reject H0
1.28
Z = .88

38. “Talk”
is not
cheap,
especially
when
you are
being
cheated !

39. Hypothesis Testing: σ Unknown
df =1
df =2
df =3
df =4
df =5
df =6

40. Hypothesis Testing: σ Unknown
Recall that the ‘t’ test statistic with n-1 degrees of freedom is:

41. Hypothesis Testing: σ Unknown Example_ _
H0: μ = H1: μ ¹ 168

42. Hypothesis Testing: σ Unknown Example
H0: μ = H1: μ ≠ 168
Determine the regions of rejection
Reject H0
α/2=.025
-t n-1,α/2
Do not reject H0
2.0639
t n-1,α/2 t
.025
24 df
2.0639

43. Hypothesis Testing: σ Unknown Example
Reject Ho
Do Not Reject Ho
Reject Ho
t n-1,α/2
-t n-1,α/2
Since t df = 24, a=.025 = 1.46 < ,
1.46
2.0639

44. Hypothesis Testing: Connection to Confidence Intervals
For X = 172.5, S = and n = 25, the 95% confidence interval is:
≤ μ ≤
Since this interval contains the hypothesized mean (168), you do not reject the null hypothesis at  = .05 _

45. Hypothesis Testing: σ Unknown

46. Hypothesis Testing: Proportions
Involves categorical variables ( yes/no , male/female )
Two possible outcomes
“Success” (possesses a certain characteristic)
“Failure” (does not possesses that characteristic)
Fraction or proportion of the population in the “success” category is denoted by π

47. Hypothesis Testing: Proportions
Sample proportion in the success category is denoted by p
When both nπ and n(1-π) are at least 5, p can be approximated
by a normal distribution with mean and standard deviation

48. Hypothesis Testing: Proportions
The sampling distribution of p is approximately normal, so the test statistic is a ‘Z’ value:

49. Hypothesis Testing: Proportions Example
A marketing company claims that it receives 8% responses from its mailings. To test this claim, a random sample of 500 were surveyed with 30 responses. Test at the  = .05 significance level.

50. Hypothesis Testing: Proportions Example
H0: π = H1: π ≠ .08
α = .05
n = 500, p = .06
Determine region of rejection
Critical Values: ± 1.96 z
Reject
.025
1.96
-1.96
30 / 500 = .06 = p
(sample proportion) Z
0.06
- 1.9
.0250
Do Not Reject Ho Z
0.06
+ 1.9
.9750

51. Hypothesis Testing: Proportions Example
Test Statistic:
Decision:
Conclusion:
.025
Do Not Reject Ho
.025 z
-1.96
1.96
-1.646

52. Potential Pitfalls and Ethical Considerations

53. Chapter Summary
In this chapter, we have