1. Chapter 8 Hypothesis Testing
STATISTICS
Chapter 8 Hypothesis Testing
C.M. Pascual

2. Chapter 8 Hypothesis Testing
8-1 Overview
8-2 Fundamentals of Hypothesis Testing
8-3 Testing a Claim about a Mean: Large     Samples
8-4 Testing a Claim about a Mean: Small     Samples
8-5 Testing a Claim about a Proportion
8-6 Testing a Claim about a Standard     Deviation

3. Definition 8-1 Overview Hypothesis
in statistics, is a claim or statement about a property of a population
page 366 of text
Various examples are provided below definition box

4. Rare Event Rule for Inferential Statistics
If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.
Example on page of text
Introduce the word ‘significant’ in regard to hypothesis testing.

5. Fundamentals of Hypothesis Testing
8-2
Fundamentals of
Hypothesis Testing

6. Figure 8-1 Central Limit Theorem
Example on page 368 of text. This is the drawing associated with that example.

7. Figure 8-1 Central Limit Theorem
The Expected Distribution of Sample Means Assuming that  = 98.6
Likely sample means
µx = 98.6

8. Figure 8-1 Central Limit Theorem
The Expected Distribution of Sample Means Assuming that  = 98.6
Likely sample means
z =
x = 98.48 or
z = 1.96
x = 98.72
µx = 98.6

9. Figure 8-1 Central Limit Theorem
The Expected Distribution of Sample Means Assuming that  = 98.6
Sample data: z =
x = 98.20
Likely sample means or
z =
x = 98.48 or
z = 1.96
x = 98.72
µx = 98.6

10. Components of a Formal Hypothesis Test
page 369 of text

11. Null Hypothesis: H0 Statement about value of population parameter
Must contain condition of equality
=, , or 
Test the Null Hypothesis directly
Reject H0 or fail to reject H0
Give examples of different wording for  and Š, such as ‘at least’, ‘at most’, ‘no more than’, etc.

12. Alternative Hypothesis: H1
Must be true if H0 is false
, <, >
‘opposite’ of Null
Give examples of different ways to word °,< and >, such as ‘is different from’, ‘fewer than’, ‘more than’, etc.

13. Note about Forming Your Own Claims (Hypotheses)
If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.
By examining the flowchart for the Wording of the Final Conclusion, Figure 7-4, page 375, this requirement for support of a statement becomes clear.

14. Note about Testing the Validity of Someone Else’s Claim
Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality).
This is important to emphasize. A claim is not always the null statement. Because of the wording, it may become the alternative.
Whichever statement the claim becomes (null or alternative), the other statement will be the ‘opposite’.
Some examples should be given for starting with a claim, and then set up the null and alternative. One example is on page 370 and exercises are appropriate.

15. Test Statistic
a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis
page 371 of text

16. For large samples, testing claims about population means
Test Statistic
a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis
For large samples, testing claims about population means
x - µx
page of text
Example on page 372 of text
z =  n

17. Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
page 372 of text

18. Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
Critical
Region

19. Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
Critical
Region

20. Critical Region
Set of all values of the test statistic that would cause a rejection of the
null hypothesis
Critical
Regions

21. Significance Level denoted by 
the probability that the test   statistic will fall in the critical   region when the null hypothesis   is actually true.
common choices are 0.05, 0.01,   and 0.10
This is the same  introduced in Section 6-2, where we defined the degree of confidence for a confidence interval to be the probability 1 - 

22. Critical Value
Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis
page 372 of text

23. Critical Value
Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis
The critical value depends on the type of test being conducted. The text will start with critical values that are z scores. Later tests will have critical values that are t scores and X2 values.
Example on page 373 of text.
Critical Value
( z score )

24. Critical Value
Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis
Reject H0
Fail to reject H0
The critical value separates the curve into areas where one would reject the null (the critical region), and where one would fail to reject the null (the rest of the curve).
Critical Value
( z score )

25. Two-tailed,Right-tailed, Left-tailed Tests
The tails in a distribution are the extreme regions bounded
by critical values.
page 373 of text

26. Two-tailed Test
H0: µ = 100
H1: µ  100

27.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region

28.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region
Means less than or greater than
Analysis of what the symbol ° means which helps students to realize this is a two tailed test.

29.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region
Means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100

30. Right-tailed Test
H0: µ  100
H1: µ > 100

31. Right-tailed Test
H0: µ  100
H1: µ > 100
Points Right

32. Right-tailed Test H0: µ  100 H1: µ > 100 Points Right Values that
Fail to reject H0
Reject H0
Values that
differ significantly
from 100
100

33. Left-tailed Test
H0: µ  100
H1: µ < 100

34. Left-tailed Test
H0: µ  100
H1: µ < 100
Points Left

35. Left-tailed Test H0: µ  100 H1: µ < 100 Points Left Values that
Reject H0
Fail to reject H0
Values that
differ significantly
from 100
100

36. Conclusions in Hypothesis Testing
always test the null hypothesis
1. Reject the H0
2. Fail to reject the H0
need to formulate correct wording of final conclusion
See Figure 8-4
page 374 of text.
Examples at bottom of page and top of page 375

37. FIGURE 8-4 Wording of Final Conclusion
Start
Does the
original claim contain
the condition of
equality
Yes
(Reject H0)
“There is sufficient
evidence to warrant
rejection of the claim
that. . . (original claim).”
(This is the
only case in
which the
original claim
is rejected).
Yes
(Original claim
contains equality
and becomes H0) Do
you reject
H0?. No
(Fail to
reject H0)
“There is not sufficient
evidence to warrant
rejection of the claim
that. . . (original claim).” No
(Original claim
does not contain
equality and
becomes H1)
(This is the
only case in
which the
original claim
is supported). Do
you reject
H0?
Yes
(Reject H0)
“The sample data
supports the claim that
(original claim).”
page 375 of text
Also found on Formula and Table insert provided with text.
Many instructors allow students to use this flowchart when taking exams.
The conclusion process indicted in this chart will be used with other statistical processes in other chapters.
Discussion of the two results that support or reject conclusions, whereas the other two do not provide enough evidence for support or rejection. No
(Fail to
reject H0)
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”

38. Accept versus Fail to Reject
some texts use “accept the null hypothesis
we are not proving the null hypothesis
sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect)
page 374 of text
The term ‘accept’ is somewhat misleading, implying incorrectly that the null has been proven. The phrase ‘fail to reject’ represents the result more correctly.

39. Type I Error
The mistake of rejecting the null hypothesis when it is true.
(alpha) is used to represent the probability of a type I error
Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6
Example on page 375 of text

40. Type II Error
the mistake of failing to reject the null hypothesis when it is false.
ß (beta) is used to represent the probability of a type II error
Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6

41. Table 8-2 Type I and Type II Errors
True State of Nature
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis) 
We decide to
reject the
null hypothesis
Correct
decision
Decision
Type II error
(rejecting a false
null hypothesis) 
We fail to
reject the
null hypothesis
Correct
decision
page 376 of text

42. Controlling Type I and Type II Errors
For any fixed , an increase in the sample size n will cause a decrease in 
For any fixed sample size n , a decrease in  will cause an increase in . Conversely, an increase in  will cause a decrease in  .
To decrease both  and , increase the sample size.
page 377 in text

43. Power of a Hypothesis Test
Definition
Power of a Hypothesis Test
is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level  and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis.

44. Steps in Hypothesis Testing
State the null and alternative hypothesis;
Select the level of significance;
Determine the critical value and the rejection region/s;
State the decision rule;
Compute the test statistics; and
Make a decision, whether to reject or not to reject the null hypothesis.

45. Example 1
A manufacturer claims that the average lifetime of his lightbulbs is 3 years or 36 months. The stabdard deviation is 8 months. Fifty (50) bulbs are selected, and the average lifetime is found to be 32 months. Should the manufacturer’s statement be rejected at = 0.01?

46. Example 1 Solution: Step 1. State the hypothesis:
Ho: µ = 36 months
Ha : µ  36 months
Step 2. Level of significance = 0.01
Step 3. Determine critical values and rejection region

47. Example 1
Solution:
Step 3. Determine critical values and rejection region
Z = +/ (from Appendix B of z values)
Step 4. State the decision rule
Reject the null hypothesis if Zc > or Zc =

48. Example 1 zc = x - µx Solution: Step 5. Compute the test statistic. 

49. Example 1 Solution: Step 6. Make a decision.
Zc = is less than Z =
And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months.

50. Example 1 Solution: Step 6. Make a decision.
Zc = is less than Z =
And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months.

51. Example 2
A test on car braking reaction times for men between 18 to 36 years old have produced a mean and standard deviation of second and second, respectively. When 40 male drivers of this age group were randomly selected and tested for their breaking reaction times, a mean of second came out. At the  = 0.10, test the claim of the driving instructor that his graduates had faster reaction times.
Zc = is less than Z =
And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months.

52. Example 2 Solution: Step 1. State the hypothesis:
Ho: µ = second
Ha: µ < second
Step 2. Level of significance = 0.10

53. Example 2
Solution:
Step 3. Determine critical values and rejection region
Z = - 1/.28 (from Appendix B of z values)
Step 4. State the decision rule
Reject the null hypothesis if Zc < .

54. Example 2 zc = x - µx Solution: Step 5. Compute the test statistic. 

55. Example 2 Solution: Step 6.
Since the test statistics falls within the non-critical region, do not reject Ho. There is enough evidence to support the instructor’s claim.; accept Ho. .

56. Test on Small Sample Mean
The t-test is a statistical test for the mean of a population and is used when the population is normally distributed, σ is unknown, and n < 30. The formula for the t-test with degrees of freedom are d.f. = n – 1 is
x - µ
t = s n

57. Example 3
In order to increase customer service, a muffler repair shop claim its mechanics can replace a muffler in 12 minutes. A time management specialist selected 6 repair jobs and found their mean time to be 11.6 minutes. The standard deviation of the sample was 2.1 minutes. A  = 0.025, is there enough evidence to conclude that the mean time in changing a muffler is less than 12 minutes?

58. Example 3 Solution State the hypothesis:
Ho: µ = 12
Ha: µ < 12
Step 2. Level of significance = 0.025
Step 3. Since and d.f. = 6 – 1 = 5, then at = , Appendix C at t-value =

59. Example 1
Solution 4. Step 4. Reject Ho if tc < Compute for the test statistic
x - µ
t = s

60. Example 3
t = (11.6 – 12)/(2.1(6)0.5 =
Step 6. Since the critical value fall within the non-critical region, do not reject Ho. Accept Ho.

61. Sit Work (Submit after class)
A diet clinic states that there is an average loss of 24 pounds for those who stay on the program for 20 weeks. The standard deviation is 5 pounds. The clinic tries a new diet, reducing salt intake to see whether that strategy will produce a greater weight loss. A group of 40 volunteers loses an average of 16.3 pounds each over 20 weeks. Should the clinic change the new diet? Use  = 0.05

62. Assignment (Submit next meeting)
2. A recent survey stated that household received an average 37 telephone calls per month. To test the claim, a researcher surveyed 29 households and found that the average number of calls was The standard deviation of the sample was 6. At = 0.05, can the claim be substantiated?