1. Types of Hypotheses Research Hypothesis Statistical Hypotheses
a statement of what the researcher believes will be the outcome of an experiment or a study
Statistical Hypotheses
a formal structure used to scientifically test the research hypothesis
Substantive Hypotheses
a statistically significant difference does not imply a material, or substantive difference

2. Example Research Hypotheses
Older workers are more loyal to a company.
Companies with more than $1 billion of assets spend a higher percentage of their annual budget on advertising than do companies with less than $1 billion of assets.
The price of scrap metal is a good indicator of the industrial production index six months later.

3. Statistical Hypotheses
Two Parts
a null hypothesis
an alternative hypothesis
Null Hypothesis – nothing new is happening; the null condition exists
Alternative Hypothesis – something new is happening
Notation
null: H0
alternative: Ha

4. Null and Alternative Hypotheses
The Null and Alternative Hypotheses are mutually exclusive. Only one of them can be true.
The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.)
The Null Hypothesis is assumed to be true.
The burden of proof falls on the Alternative Hypothesis. 5

5. Null and Alternative Hypotheses: Example
A manufacturer is filling 40 oz. packages with flour.
The company wants the package contents to average 40 ounces. 6

6. Null and Alternative Hypotheses: Example
Because of an increase marketing effort, company officials believe the company’s market share is now greater than 18%, and the officials would like to prove it. 6

7. HTAB System to Test Hypotheses
Task 1:
HYPOTHESIZE
Task 2:
TEST
Task 3:
TAKE STATISTICAL ACTION
Task 4:
DETERMINING THE
BUSINESS IMPLICATIONS

8. Steps in Testing Hypotheses
1. Establish hypotheses: state the null and alternative hypotheses.
2. Determine the appropriate statistical test and sampling distribution.
3. Specify the Type I error rate (
4. State the decision rule.
5. Gather sample data.
6. Calculate the value of the test statistic.
7. State the statistical conclusion.
8. Make a managerial decision. 4

9. HTAB Paradigm – Task 1 Task 1: Hypotheses
Step 1. Establish hypotheses: state the null and alternative hypotheses. 4

10. HTAB Paradigm – Task 2 Task 2: Test
Step 2. Determine the appropriate statistical test and sampling distribution.
Step 3. Specify the Type I error rate (
Step 4. State the decision rule.
Step 5. Gather sample data.
Step 6. Calculate the value of the test statistic. 4

11. HTAB Paradigm – Task 3 Task 3: Take Statistical Action
Step 7. State the statistical conclusion. 4

12. HTAB Paradigm – Task 4 Task 4: Determine the business implications
Step 8. Make a managerial decision. 4

13. Rejection and Nonrejection Regions
Using the critical values established at Step 4 of the hypothesis testing process, the possible statistical outcomes of a study can be divided into two groups:
Those that cause the rejection of the null hypothesis
Those that do not cause the rejection of the null hypothesis

14. Rejection and Nonrejection Regions
Conceptually and graphically, statistical outcomes that result in the rejection of the null hypothesis lie in what is termed the rejection region.
Statistical outcomes that fail to result in the rejection of the null hypothesis lie in what is termed the nonrejection region.

15. Possible Rejection and Nonrejection Regions -
There are three possibilities which can be stipulated in the alternative hypothesis.
The three possibilities are: >, <, or .
The rejection regions for these possibilities, if a standard normal distribution is used for the test statistic, are shown on the next slide.

16. Possible Rejection and Nonrejection Regions -
for hypothesis
which involve the
standard normal
distribution and
the > symbol (right –tailed test)

17. Possible Rejection and Nonrejection Regions -
for hypothesis
which involve the
standard normal
distribution and
the < symbol (left –tailed test)

18. Possible Rejection and Nonrejection Regions -
for hypothesis
which involve the
standard normal
distribution and
the  symbol (two –tailed test)

19. Type I and Type II Errors
Type I Error
Rejecting a true null hypothesis
The probability of committing a Type I error is called , the level of significance.
Type II Error
Failing to reject a false null hypothesis
The probability of committing a Type II error is called . 8

20. Decision Table for Hypothesis Testing( )
Null True
Null False
Fail to
reject null
Correct
Decision
Type II error 
Reject null
Type I error 
Correct Decision 9

21. Testing Hypotheses about a Population Mean Using the z Statistic ( Known)
Example: A survey, done 10 years ago, of CPAs in the U.S. found that their average salary was $74,914. An accounting researcher would like to test whether this average has changed over the years. A sample of 112 CPAs produced a mean salary of $78,695. Assume that the population standard deviation of salaries  = $14,530.

22. Testing Hypotheses about a Population Mean Using the z Statistic ( Known)
Step 1: Hypothesize
Step 2: Test

23. Testing Hypotheses about a Population Mean Using the z Statistic ( Known)
Step 3: Specify the Type I error rate-
 =  z/2 = 1.96
Step 4: Establish the decision rule-
Reject H0 if the test statistic < or it the test statistic > 1.96.

24. Testing Hypotheses about a Population Mean Using the z Statistic ( Known)
Step 5: Gather sample data-
x-bar = $78,695, n = 112,  = $14,530, hypothesized  = $74,914.
Step 6: Compute the test statistic.

25. Testing Hypotheses about a Population Mean Using the z Statistic ( Known)
Step 7: Reach a statistical conclusion-
Since z = 2.75 > 1.96, reject H0.
Step 8: Business decision-
Statistically, the researcher has enough evidence to reject the figure of $74,914 as the true average salary for CPAs. In addition, based on the evidence gathered, it may suggest that the average has increased over the 10-year period.

26. Testing Hypotheses about a Population Mean Using the z Statistic ( Known) from a Finite Population
Test statistic:

27. Using the p-Value to Test Hypotheses
Another way to reach a statistical conclusion in hypothesis testing problems is by using the p-value, sometimes referred to as the observed significance level.
p-value <   reject H0
p-value    do not reject H0

28. Using the p-Value to Test Hypotheses
One should be careful when using p-values from statistical software outputs.
Both MINITAB and EXCEL report the actual p-values for hypothesis tests.
MINITAB doubles the p-value for a two-tailed test so you can compare with .
EXCEL does not double the p-value for a two-tailed test. So when using the p-value from EXCEL, you may multiply the value by 2 and then compare with .

29. Demonstration Problem: MINITAB20

30. Using the p-Value to Test Hypotheses

31. Critical Value Method to Test Hypotheses
The critical value method determines the critical mean value required for z to be in the rejection region and uses it to test the hypotheses.

32. Critical Value Method to Test Hypotheses
For the previous example,

33. Critical Value Method to Test Hypotheses
Thus, a sample mean greater than $77,605 or less than $72,223 will result in the rejection of the null hypothesis.
This method is particularly attractive in industrial settings where standards can be set ahead of time and then quality control technicians can gather data and compare actual measurements of products to specifications.

34. Testing Hypotheses about a Population Mean Using the t Statistic ( Unknown)
In this case, the test statistic will be

35. Two-tailed Test:  Unknown,  = .05 (Part 1)
Example: Weights in Pounds of a Sample of 20 Plates 21

36. MINITAB Computer Printout for the Machine Plate Example
Ha:   25
Do not reject the null hypothesis
since P-value = >  = 0.05. 24

37. Machine Plate Example: Excel (Part 1)
Do not reject the
null hypothesis
since
P-value = >  = 0.05.

38. Machine Plate Example: Excel (Part 2)B C D E 1
H0: m = 25 2
Ha: m ¹ 3 4
22.6
22.2
23.2
27.4
24.5 5 27
26.6
28.1
26.9
24.9 6
26.2
25.3
23.1
24.2
26.1 7
25.8
30.4
28.6
23.5
23.6 8 9
n =
=COUNT(A4:E7) 10
a =
0.05 11
Mean =
=AVERAGE(A4:E7) 12
S =
=STDEV(A4:E7) 13
Std Error =
=B12/SQRT(B9) 14
t =
=(B11-B1)/B13 15
p-Value
=TDIST(B14,B9-1,2)

39. z Test of Population Proportion28

40. z Test of Population Proportion
A manufacturer believes exactly 8% of its products contain at least one minor flaw. Suppose the company wants to test this belief. A sample of 200 products resulted in 33 items have at least one minor flaw. Use a probability of a Type I error of 0.10.
H0: p = 0.08
Ha: p  0.08

41. Testing Hypotheses about a Proportion: Manufacturer Example (Part 2)
Critical Values
Non Rejection Region
Rejection Regions 30

42. MINITAB Computer Printout for the Minor Flaw Example
H0: p = 0.08
Ha: p  0.08
Reject the null hypothesis
since P-value = <  = 0.1. 24

43. Using the Critical Value Method
H0: p = 0.08
Ha: p  0.08
Since the sample
proportion of 0.165
falls outside the
interval, the null
hypothesis
is rejected 24

44. Testing Hypotheses About a Variance
The test statistic for this test is

45. Testing Hypotheses About a Variance: Demonstration Problem 9.4
Step 1:
Step 2: Test statistic
H0: 2 = 25
Ha: 2  25

46. Testing Hypotheses About a Variance: Demonstration Problem 9.4
Step 3: Because this is a two-tailed test,  = 0.10 and /2 = 0.05.
Step 4: The degrees of freedom are 16 – 1 = 15. The two critical chi-square values are 2(1 – 0.05), 15 = 2 0.95, 15 = and 2 0.05, 15 =
Step 5: The data are listed in the text.
Step 6: The sample variance is s2 = The observed chi-square value is calculated as 2 =

47. Testing Hypotheses About a Variance: Demonstration Problem 9.4
Step 7: The observed chi-square value is in the nonrejection region because 2 0.95, 15 = < 2observed = < 2 0.05), 15 =
Step 8: This result indicates to the company managers that the variance of weekly overtime hours is about what they expected.

48. Solving for Type II Errors
When the null hypothesis is not rejected, then either a correct decision is made or an incorrect decision is made.
If an incorrect decision is made, that is, if the null hypothesis is not rejected when it is false, then a Type II error has occurred.
Finding the probability of a Type II error is more complex than finding the probability of Type I error.
A Type II error, , varies with possible values of the alternative parameter.

49. Solving for Type II Errors (Soft Drink)
Suppose a test is conducted on the following hypotheses: H0: = 12 ounces vs. Ha: < 12 ounces when the sample size is 60 with mean of
The first step in determining the probability of a Type II error is to calculate a critical value for the sample mean (in this case).
For an  =0.05, then the critical value for the sample mean is (given on next slide).

50. Solving for Type II Errors (Soft Drink)
In testing the null hypothesis
by the critical value method,
this value is used as the cutoff
for the nonrejection region. For any sample mean obtained that is less than , the null hypothesis is rejected. Any sample mean greater than , the null hypothesis is not rejected.

51. Solving for Type II Errors (Soft Drink)
Since a Type II error, , varies with possible values of the alternative parameter, then for an alternative mean of (< 12) the corresponding z-value is

52. Solving for Type II Errors (Soft Drink)
INSERT FIGURE 9.20

53. Solving for Type II Errors (Soft Drink)
The value of z yields an area of
The probability of committing a Type II error is equal to the area to the right of the critical value of the sample mean of
This area is = =
Thus, there is an 80.23% chance of committing a Type II error if the alternative mean is
Note: equivalent problems can be solved for sample proportions (See Demonstration Problem 9.6).

54. Operating Characteristic and Power Curve
Because the probability of committing a Type II error changes for each different value of the alternative parameter, it is best to examine a series of possible alternative values.
The power of a test is the probability of rejecting the null hypothesis when it is false.
Power = 1 - .

55. Operating Characteristic and Power Curve (Soft Drink)

56. Operating Characteristic and Power Curve (Soft Drink)

57. Operating Characteristic and Power Curve (Soft Drink)

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