1. Overview Definition 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

2. 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.

3. Alternative Hypothesis: Ha
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.

4. 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.

5. HO The defendant Claim about a
Legal Trial
Hypothesis Test
HO The defendant Claim about a
is not guilty population parameter
HA The defendant Opposing claim about a
is guilty population parameter
Result The evidence The statistic indicates a
convinces the rejection of HO, and the
jury to reject alternate hypothesis is
the assumption accepted.
of innocence. The
verdict is guilty
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.

6. 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

7. 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
page of text
Example on page 372 of text
x - µx
z =  n

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

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

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

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

12. 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 - 

13. 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

14. 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 )

15. 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 )

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

17. Two-tailed Test
H0: µ = 100
Ha: µ  100

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

19.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
Ha: 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.

20.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
Ha: 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

21. Right-tailed Test
H0: µ  100
Ha: µ > 100

22. Right-tailed Test
H0: µ  100
Ha: µ > 100
Points Right

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

24. Left-tailed Test
H0: µ  100
Ha: µ < 100

25. Left-tailed Test
H0: µ  100
Ha: µ < 100
Points Left

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

27. 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
page 374 of text.
Examples at bottom of page and top of page 375

28. Wording of Final Conclusion
Start
Does the
original claim contain
the condition of
equality
“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?.
Yes
(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 Ha)
(This is the
only case in
which the
original claim
is supported).
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. Do
you reject
H0?
Yes
(Reject H0)
“The sample data
supports the claim that
(original claim).” No
(Fail to
reject H0)
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”

29. 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.

30. 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.

31. for testing claims about population means
Assumptions
for testing claims about population means
1) The sample is a simple random       sample.
2) The sample is large (n > 30).
a) Central limit theorem applies
b) Can use normal distribution
3) If  is unknown, we can use sample standard deviation s as estimate for .
page 381 of text

32. Traditional (or Classical) Method of Testing Hypotheses
Goal
Identify a sample result that is significantly different from the claimed value
page 382 of text

33. The traditional (or classical) method of hypothesis testing converts the relevant sample statistic into a test statistic which we compare to the critical value.

34. Hypotheses Testing 5 Step Process
1. State the hypotheses
2. Decide on a model.
3. Determine the endpoints of the rejection region and state the decision rule.
4. Compute the test statistic
5. State the conclusion
These are the steps in the flowchart of Figure 7-5 on page 383.

35. Test Statistic for Claims about µ when n > 30
x - µx
z = 
Test Statistic for Claims about µ when n < 30
x - µx
t =  n

36. Decision Criterion
Reject the null hypothesis if the test statistic is in the critical region
Fail to reject the null hypothesis if the test statistic is not in the critical region

37. FIGURE 7-4 Wording of Final Conclusion
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 Ha)
(This is the
only case in
which the
original claim
is supported).
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. Do
you reject
H0?
Yes
(Reject H0)
“The sample data
supports the claim that
(original claim).” No
(Fail to
reject H0)
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”

38. Example: Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o.
This examples starts at the bottom of page 383 and concludes on page
Remind students that the first time through the entire Hypothesis Testing procedure will seem ‘intense’. However, after many practices (doing some exercises provided in the text), the process will begin to seem a little more ‘natural’.

39. Example: Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o.
Steps:
1) State the hypotheses
H0 :  = 98.6o
Ha :   98.6o
2) Determine the model
Setting up the claim, null, and alternative are critical to the rest of the process. If any of these are identified incorrectly, the final result of the procedure will be in great jeopardy.
Two tail Z test, n > 30

40.  = 0.05 /2 = 0.025 (two tailed test) z = - 1.96 1.96
3) Determine the Rejection Region
 = 0.05
/2 = (two tailed test)
0.4750
0.4750
0.025
0.025
This is a two-tailed test because the alternative indicated ‘is not equal to’. See Section 7-2, page
The critical values are determined from Table A-2, looking up the area of in the body of the table. (Review Section 5-4, page 249 of text.
z =

41. z = = = - 6.64 x - µ 98.2 - 98.6  4) Compute the test statistic n
0.62 n
106
Care must be taken when using a calculator to find the z value. If students wish to compute the value all in one step on the calculator, parentheses will need to be placed around the numerator and the denominator.

42. 5) State the Conclusion z = - 6.64 REJECT H0 z = - 1.96 µ = 98.6
Sample data: x = 98.2o or
z =
Reject
H0: µ = 98.6
Reject
H0: µ = 98.6
Fail to Reject
H0: µ = 98.6
Place the test statistic z score (-6.64) correctly in place on the z score number line. This correct placement puts it far down into the left critical region.
z =
µ = 98.6
or z = 0
z = 1.96
z =
There is sufficient evidence to warrant rejection of claim that
the mean body temperatures of healthy adults is equal to 98.6o.
REJECT H0

43. for testing claims about population means
Assumptions
for testing claims about population means
1) The sample is a simple random sample.
2) The sample is small (n  30).
3) The value of the population standard  deviation  is unknown.
4) The sample values come from a population  with a distribution that is approximately  normal.
page 399 of text

44. Test Statistic for a Student t-distribution
x -µx
t = s n
Critical Values
Found in Table A-3
Degrees of freedom (df) = n -1
Critical t values to the left of the mean are negative
page 400 of text
Most common mistake made with this procedure is to not use Table A-3 to find the critical values. Some use Table A-2 incorrectly.

45. Important Properties of the Student t Distribution
1. The Student t distribution is different for different sample sizes (see Figure 6-5 in Section 6-3).
2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected with small samples.
3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a = 1).
5. As the sample size n gets larger, the Student t distribution get closer to the normal distribution. For values of n > 30, the differences are so small that we can use the critical z values instead of developing a much larger table of critical t values. (The values in the bottom row of Table A-3 are equal to the corresponding critical z values from the normal distributions.)
page 400 of text

46. the population essentially
Figure Choosing between the Normal and Student t-Distributions when Testing a Claim about a Population Mean µ
Start
Use normal distribution with
x - µx Is
n > 30 ?
Yes
Z
/ n
(If  is unknown use s instead.) No
Is the
distribution of
the population essentially
normal ? (Use a
histogram.) No
Use nonparametric methods, which don’t require a normal distribution.
Yes
Use normal distribution with
page 401 of text
Example of the small sample procedure is on page 402 of text
Is 
known ?
x - µx
Z
/ n No
(This case is rare.)
Use the Student t distribution
with
x - µx
t
s/ n

47. The larger Student t critical value shows that with a small sample, the sample evidence must be more extreme before we consider the difference is significant.
bottom of page 403 of text

48. A company manufacturing rockets claims to use an
average of 5500 lbs of rocket fuel for the first 15 seconds of
operation. A sample of 6 engines are fired and the mean fuel
consumption is 5690 lbs with a sample standard deviation of
250 lbs. Is the claim justified at the 5% level of significance?
2.571
-2.571
1. HO: µ = HA: µ  5500
2. Two tail t test, n < 30, unknown
population standard deviation
1.862
3. t critical for 5% for a two tail test with 5 d.f. is 2.571
Fail to reject HO, there is no evidence at the .05 level
that the average fuel consumption is different from µ = 5500 lbs

49. P-Value Method Table A-3 includes only selected values of 
Specific P-values usually cannot be found
Use Table to identify limits that contain the P-value
Some calculators and computer programs will find exact P-values
page 404 of text
Example on this page too.

50. P-Value Method of Testing Hypotheses
very similar to traditional method
key difference is the way in which we decide to reject the null hypothesis
approach finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low
page 387 of text

51. P-Value Method of Testing Hypotheses
Definition
P-Value (or probability value)
the probability that the test statistic is as far or farther from  if the null hypothesis is true
The attained significance level of a hypothesis test is the P value of its test statistic

52. P-Value Method of Testing Hypotheses
Guidelines for rejecting HO based on the P value:
If P < , then reject HO and accept HA
If P > , then reserve judgement about HO

53. P-value Interpretation Small P-values (such as 0.05 or lower)
Unusual sample results. Significant difference from the null hypothesis
Large P-values (such as above 0.05)
Sample results are not unusual. Not a significant difference from the null hypothesis

54. Figure 7-8 Finding P-Values
Start
Left-tailed
What
type of test ?
Right-tailed
Two-tailed Is
the test statistic
to the right or left of
center ?
Left
Right
P-value = area
to the left of the
test statistic
page 390 of text
P-value = twice
the area to the left
of the test statistic
P-value = twice
the area to the right
of the test statistic
P-value = area
to the right of the
test statistic
P-value
P-value is twice
this area
P-value is twice
this area
P-value
Test statistic
Test statistic
Test statistic
Test statistic

55. Testing Claims with Confidence Intervals
A confidence interval estimate of a population parameter contains the likely values of that parameter. We should therefore reject a claim that the population parameter has a value that is not included in the confidence interval.
page 392 of text

56. Testing Claims with Confidence Intervals
Claim: mean body temperature = 98.6º,
where n = 106, x = 98.2º and s = 0.62º
95% confidence interval of 106 body temperature data (that is, 95% of samples would contain true value µ )
98.08º < µ < 98.32º
98.6º is not in this interval
Therefore it is very unlikely that µ = 98.6º
Thus we reject claim µ = 98.6º

57. Underlying Rationale of Hypotheses Testing
If, under a given observed assumption, the probability of getting the sample is exceptionally small, we conclude that the assumption is probably not correct.
When testing a claim, we make an assumption (null hypothesis) that contains equality. We then compare the assumption and the sample results and we form one of the following conclusions:
page 393 of text

58. Underlying Rationale of Hypotheses Testing
If the sample results can easily occur when the assumption (null hypothesis) is true, we attribute the relatively small discrepancy between the assumption and the sample results to chance.
If the sample results cannot easily occur when that assumption (null hypothesis) is true, we explain the relatively large discrepancy between the assumption and the sample by concluding that the assumption is not true.

59. 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

60. 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

61. 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) 
page 376 of text
We fail to
reject the
null hypothesis
Correct
decision

62. 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