1. Statistics for Managers Using Microsoft Excel
Fundamentals of Hypothesis Testing Chapter 7
Learning Objectives
Describe the hypothesis testing process
Distinguish the types of hypotheses
Explain hypothesis testing errors
Solve hypothesis testing problems
One population mean
One population proportion
One & two-tailed tests
As a result of this class, you will be able to...

2. Statistical Methods

3. Reject hypothesis! Not close.
Hypothesis Testing
I believe the population mean age is 50 (hypothesis).
Reject hypothesis! Not close.
Population      
Random sample 
Mean X = 20  

4. What’s a Hypothesis? A belief about a population parameter
Parameter is population mean, proportion, variance
Must be stated before analysis
I believe the mean GPA of this class is 3.5!
© T/Maker Co.

5. Null Hypothesis What is tested
Has serious outcome if incorrect decision made
Always has equality sign: , or 
Designated H0
Pronounced ‘H sub-zero’ or ‘H oh’
Example
H0:   3

6. Alternative Hypothesis
Opposite of null hypothesis
Always has inequality sign: ,, or 
Designated H1
Example
H1:  < 3

7. Identifying Hypotheses in Problems
Problem: Test that the population mean is not 3
Steps
State the question statistically:   3
State the opposite statistically:  = 3
Must be mutually exclusive & exhaustive
Select the null hypothesis:  = 3
Has the =, , or sign

8. Identifying Hypotheses Thinking Challenge
What are the hypotheses?
Is the population average amount of TV viewing 12 hours?

9. Identifying Hypotheses Thinking Challenge
What are the hypotheses?
Is the population average amount of TV viewing 12 hours?
Is the population average amount of TV viewing different from 12 hours?

10. Identifying Hypotheses Thinking Challenge
What are the hypotheses?
Is the population average amount of TV viewing 12 hours?
Is the population average amount of TV viewing different from 12 hours?
Is the average cost per hat less than or equal to $20?

11. Identifying Hypotheses Thinking Challenge
What are the hypotheses?
Is the population average amount of TV viewing 12 hours?
Is the population average amount of TV viewing different from 12 hours?
Is the average cost per hat less than or equal to $20?
Is the average amount spent in the bookstore greater than $25?

12. Sampling Distribution
Basic Idea
Sampling Distribution
It is unlikely that we would get a sample mean of this value ...
... therefore, we reject the hypothesis that  = 50.
... if in fact this were the population mean 20 H0

13. Level of Significance
Defines unlikely values of sample statistic if null hypothesis is true
Called rejection region of sampling distribution
Designated (alpha)
Typical values are .01, .05, .10
Selected by researcher at start

14. Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
1 - 
Rejection region does NOT include critical value.
Observed sample statistic

15. Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
1 - 
Rejection region does NOT include critical value.
Observed sample statistic

16. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
1 - 
Rejection region does NOT include critical value.
Observed sample statistic

17. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
1 - 
Rejection region does NOT include critical value.
Observed sample statistic

18. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
1 - 
Rejection region does NOT include critical value.
Observed sample statistic

19. Risk of Errors in Making Decision
Type I error
Reject true null hypothesis
Has serious consequences
Probability of Type I error is 
Called level of significance
Type II error
Do not reject false null hypothesis
Probability of Type II error is (Beta)

20. Decision Results
H0: Innocent

21.  &  Have an Inverse Relationship
You can’t reduce both errors simultaneously!  

22.  increases when difference with hypothesized parameter decreases
Factors Affecting 
True value of population parameter
 increases when difference with hypothesized parameter decreases
Significance level, 
Increases when decreases
Population standard deviation, 
Increases when  increases
Sample size, n
Increases when n decreases

23. Hypothesis H0 Testing Steps
State H0
State H1
Choose 
Choose n
Choose test

24. H0 Testing Steps State H0 Set up critical values State H1 Collect data
Choose 
Choose n
Choose test
Set up critical values
Collect data
Compute test statistic
Make statistical decision
Express decision

25. One Population Tests

26. Two-Tailed Z Test for Mean ( Known)
Assumptions
Population is normally distributed
If not normal, can be approximated by normal distribution (n  30)
Null hypothesis has = sign only
Z-test statistic

27. Two-Tailed Z Test Example
Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed X = The company has specified  to be 15 grams. Test at the .05 level.
368 gm.

28. Two-Tailed Z Test Solution
H0:  = 368
H1:   368
  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is not 368

29. p-Value
Probability of obtaining a test statistic more extreme (or than actual sample value given H0 is true
Called observed level of significance
Smallest value of  H0 can be rejected
Used to make rejection decision
If p-value  , do not reject H0
If p-value < , reject H0

30. Two-Tailed Z Test p-Value Solution
p-value is P(Z  or Z  1.50) = .1336 
.4332 
From Z table: lookup 1.50 
Z value of sample statistic

31. Two-Tailed Z Test p-Value Solution
Do not reject.
1/2 p-Value = .0668
1/2 p-Value = .0668
1/2  = .025
1/2  = .025
Test statistic is in ‘Do not reject’ region

32. Two-Tailed Z Test Thinking Challenge
You’re a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with  = 3.5 lb. You take a sample of 36 cords & compute a sample mean of 69.7 lb. At the .05 level, is there evidence that the machine is not meeting the average breaking strength?

33. Two-Tailed Z Test Solution*
H0:  = 70
H1:   70
 = .05
n = 36
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is not 70

34. One-Tailed Z Test for Mean ( Known)
Assumptions
Population is normally distributed
If not normal, can be approximated by normal distribution (n  30)
Null hypothesis has  or  sign only

35. One-Tailed Z Test for Mean ( Known)
Assumptions
Population is normally distributed
If not normal, can be approximated by normal distribution (n  30)
Null hypothesis has  or  sign only
Z-test statistic

36. One-Tailed Z Test for Mean Hypotheses
H0:0 H1: < 0
H0:0 H1: > 0
Must be significantly below 
Small values satisfy H0 . Don’t reject!

37. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025? 
 = .025 

38. One-Tailed Z Test Finding Critical Z
What Is Z given  = .025?
Standardized Normal Probability Table (Portion)  
.06
 = .025  
1.9
.4750

39. One-Tailed Z Test Example
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showedX = The company has specified  to be 15 grams. Test at the .05 level.
368 gm.

40. One-Tailed Z Test Solution
H0:   368
H1:  > 368
 = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
No evidence average is more than 368

41. One-Tailed Z Test p-Value Solution
Do not reject.
p-Value = .0668
 = .05
Test statistic is in ‘Do not reject’ region

42. One-Tailed Z Test Thinking Challenge
You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3.8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30.7 mpg. At the .01 level, is there evidence that the miles per gallon is at least 32?

43. Solution Template H0: H1:  = n = Critical Value(s): Test Statistic:
Decision:
Conclusion:

44. One-Tailed Z Test Solution*
H0:
H1:
=
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

45. One-Tailed Z Test Solution*
H0:   32
H1:  < 32
=
n =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

46. One-Tailed Z Test Solution*
H0:   32
H1:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

47. One-Tailed Z Test Solution*
H0:   32
H1:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

48. One-Tailed Z Test Solution*
H0:  32
H1:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

49. One-Tailed Z Test Solution*
H0:   32
H1:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01

50. One-Tailed Z Test Solution*
H0:   32
H1:  < 32
 = .01
n = 60
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .01
There is evidence average is less than 32

51. p-Value Thinking Challenge
You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3.8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30.7 mpg. What is the value of the observed level of significance (p-Value)? .

52.     p-Value Solution*
p-Value is P(Z  -2.65) = p-Value < ( = .01). Reject H0.  
Use alternative hypothesis to find direction
.4960 
Z value of sample statistic 
From Z table: lookup 2.65

53. One Population Tests

54. t Test for Mean ( Unknown)
Assumptions
Population is normally distributed
If not normal, only slightly skewed & large sample (n  30) taken
Parametric test procedure
t test statistic

55. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10 
df = n - 1 = 2 
 /2 = .05
 /2 = .05

56. Two-Tailed t Test Finding Critical t Values
Given: n = 3;  = .10
Critical Values of t Table (Portion)  
df = n - 1 = 2 
 /2 = .05
2.920
 /2 = .05 

57. Two-Tailed t Test Example
Does an average box of cereal contain 368 grams of cereal? A random sample of 36 boxes had a mean of & a standard deviation of 12 grams. Test at the .05 level.
368 gm.

58. Two-Tailed t Test Solution
H0:  = 368
H1:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

59. Two-Tailed t Test Solution
H0:  = 368
H1:   368
 = .05
df = = 35
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence pop. average is not 368

60. Two-Tailed t Test Thinking Challenge
You work for the FTC. A manufacturer of detergent claims that the mean weight of detergent is 3.25 lb. You take a random sample of 64 containers. You calculate the sample average to be lb. with a standard deviation of .117 lb. At the .01 level, is the manufacturer correct?
Allow students about 10 minutes to finish this.
3.25 lb.

61. Two-Tailed t Test Solution*
H0:  = 3.25
H1:   3.25
  .01
df  = 63
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .01
There is no evidence average is not 3.25

62. One-Tailed t Test Example
Is the average capacity of batteries at least 140 ampere-hours? A random sample of 20 batteries had a mean of & a standard deviation of Assume a normal distribution. Test at the .05 level.

63. One-Tailed t Test Solution
H0:   140
H1:  < 140
 = .05
df = = 19
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence pop. average is less than 140

64. One-Tailed t Test Thinking Challenge
You’re a marketing analyst for Wal-Mart. Wal-Mart had teddy bears on sale last week. The weekly sales ($ 00) of bears sold in 10 stores was: At the .05 level, is there evidence that the average bear sales per store is more than 5 ($ 00)?
Assume that the population is normally distributed.
Allow students about 10 minutes to solve this.

65. One-Tailed t Test Solution*
H0:   5
H1:  > 5
 = .05
df = = 9
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
Do not reject at  = .05
There is no evidence average is more than 5

66. Data Types

67. Categorical Data
Categorical random variables yield responses that classify
e.g., gender (male, female)
Measurement reflects # in category
Nominal or ordinal scale
Examples
Do you own savings bonds?
Do you live on-campus or off-campus?

68. Proportions Involve categorical variables
Fraction or % of population in a category
If two categorical outcomes, binomial distribution
Possess or don’t possess characteristic

69. Proportions Involve categorical variables
Fraction or % of population in a category
If two categorical outcomes, binomial distribution
Possess or don’t possess characteristic
Sample proportion (ps)

70. One Population Tests

71. One-Sample Z Test for Proportion
Assumptions
Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used
n·p  5 & n·(1 - p)  5

72. One-Sample Z Test for Proportion
Assumptions
Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used
n·p  5 & n·(1 - p)  5
Z-test statistic for proportion
Hypothesized population proportion

73. One-Proportion Z Test Example
The present packaging system produces 10% defective cereal boxes. Using a new system, a random sample of 200 boxes had11 defects. Does the new system produce fewer defects? Test at the .05 level. 
n·p  5
n·(1 - p)  5

74. One-Proportion Z Test Solution
H0: p  .10
H1: p < .10
 = .05
n = 200
Critical Value(s):
Test Statistic:
Decision:
Conclusion:

75. One-Proportion Z Test Solution
H0: p  .10
H1: p < .10
 = .05
n = 200
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at  = .05
There is evidence new system < 10% defective

76. One-Proportion Z Test Thinking Challenge
You’re an accounting manager. A year-end audit showed 4% of transactions had errors. You implement new procedures. A random sample of 500 transactions had 25 errors. Has the proportion of incorrect transactions changed at the .05 level?
n·p  5
n·(1 - p)  5

77. One-Proportion Z Test Solution*
H0: p = .04
H1: p  .04
 = .05
n = 500
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at  = .05
There is evidence proportion is still 4%

78. Conclusion Described the hypothesis testing process
Distinguished the types of hypotheses
Explained hypothesis testing errors
Solved hypothesis testing problems
One population mean
One population proportion
One & two-tailed tests
As a result of this class, you will be able to ...