1. Chapter Nine Part 1 (Sections 9.1 & 9.2) Hypothesis Testing
Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College
Chapter Nine Part 1
(Sections 9.1 & 9.2)
Hypothesis Testing

2. Hypothesis testing is used to make decisions concerning the value of a parameter.

3. a working hypothesis about the population parameter in question
Null Hypothesis: H0
a working hypothesis about the population parameter in question

4. The value specified in the null hypothesis is often:
a historical value
a claim
a production specification

5. Alternate Hypothesis: H1
any hypothesis that differs from the null hypothesis

6. An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.

7. Determine the null and alternate hypotheses.
A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long.
Determine the null and alternate hypotheses.

8. A manufacturer claims that their light bulbs burn for an average of 1000 hours. ...
The null hypothesis (the claim) is that the true average life is 1000 hours.
H0:  = 1000

9. … A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. ...
If we reject the manufacturer’s claim, we must accept the alternate hypothesis that the light bulbs do not last as long as 1000 hours.
H1:  < 1000

10. rejecting a null hypothesis which is, in fact, true
Type I Error
rejecting a null hypothesis which is, in fact, true

11. not rejecting a null hypothesis which is, in fact, false
Type II Error
not rejecting a null hypothesis which is, in fact, false

12. Options in Hypothesis Testing
Our Choices:
H0 is
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13. Errors in Hypothesis Testing
Our Choices:
Type I error
H0 is

14. Errors in Hypothesis Testing
Our Choices:
Type I error
H0 is
Type II error

15. Errors in Hypothesis Testing
Our Choices:
Correct decision
Type I error
H0 is
Correct decision
Type II error

16. Level of Significance, Alpha ()
the probability with which we are willing to risk a type I error

17. Type II Error
 = beta =probability of a type II error (failing to reject a false hypothesis)
A small  is normally is associated with a (relatively) large , and vice-versa.
Choices should be made according to which error is more serious.
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18. Power of the Test = 1 – Beta
The probability of rejecting H0 when it is in fact false = 1 – .
The power of the test increases as the level of significance () increases.
Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

19. Probabilities Associated with a Hypothesis Test

20. Probabilities Associated with a Hypothesis Test

21. Probabilities Associated with a Hypothesis Test

22. Probabilities Associated with a Hypothesis Test

23. Probabilities Associated with a Hypothesis Test

24. Reject or ...
When the sample evidence is not strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis.
Use of the term “accept the null hypothesis” should be avoided.
When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

25. Fail to Reject H0
There is not enough evidence to reject H0. The null hypothesis is retained but has not been proven.

26. Reject H0
There is enough evidence to reject H0. Choose the alternate hypothesis with the understanding that it has not been proven.

27. A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than We wish to test the claim with a level of significance of  = 0.01,

28. Describe Type I and Type II errors.
… average age of its job applicants is fifteen years. We suspect that the true age is lower than 15.
H0:  = 15
H1:  < 15
Describe Type I and Type II errors.

29. H0:  = 15 H1:  < 15  = 0.01
A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

30. H0:  = 15 H1:  < 15 = 0.01
A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.

31. Types of Tests
When the alternate hypothesis contains the “not equal to” symbol (  ), perform a two-tailed test.
When the alternate hypothesis contains the “greater than” symbol ( > ), perform a right-tailed test.
When the alternate hypothesis contains the “less than” symbol ( < ), perform a left-tailed test.
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32. Two-Tailed Test
H0:  = k
H1:   k
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33. Two-Tailed Test H0:  = k H1:   k
If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis
– z z
If test statistic is in either tail - the critical region - of the distribution, we reject the Null Hypothesis.
143

34. Right-Tailed Test
H0:  = k
H1:  > k
144

35. Right-Tailed Test H0:  = k H1:  > k
If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis z
If test statistic is in the right tail - the critical region - of the distribution, we reject the Null Hypothesis.
145

36. Left-Tailed Test H0:  = k H1:  < k
If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis z
If test statistic is in the left tail - the critical region - of the distribution, we reject the Null Hypothesis.
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37. Procedure for Hypothesis Testing
1. Establish the null hypothesis, H0.
2. Establish the alternate hypothesis: H1.
3. Use the level of significance and the alternate hypothesis to determine the critical region.
4. Find the critical values that form the boundaries of the critical region(s).
5. Use the sample evidence to draw a conclusion regarding whether or not to reject the null hypothesis.
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38. Claim about  or historical value of  H0:  = k
Null Hypothesis
Claim about  or historical value of 
H0:  = k

39. If you believe  is less than the value stated in H0,
H0:  = k
If you believe  is less than the value stated in H0,
use a left-tailed test.
H1:  < k

40. H0:  = k If you believe  is more than the value stated in H0,
use a right-tailed test.
H1:  > k

41. If you believe  is different from the value stated in H0,
H0:  = k
If you believe  is different from the value stated in H0,
use a two-tailed test.
H1:   k

42. Apply the Central Limit Theorem.
Hypothesis Testing About a Population Mean  when Sample Evidence Comes From a Large Sample
Apply the Central Limit Theorem.

43. Central Limit Theorem Indicates:
Since we are working with assumptions concerning a population mean for a large sample, we can assume:
1. The distribution of sample means is (approximately) normal.
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44. Central Limit Theorem Indicates:
Since we are working with assumptions concerning a population mean for a large sample, we can assume:
2. The mean of the sampling distribution is the same as the mean of the original distribution.
136

45. Assumptions:
Since we are working with assumptions concerning a population mean for a large sample, we can assume:
3. The standard deviation of the sampling distribution = the original standard deviation divided by the square root of the the sample size.
137

46. Use of the Level of Significance
For a one-tailed test,  is the area in the tail (the rejection area). 
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47. Use of the Level of Significance
For a two-tailed test,  is the total area in the two tails.
Each tail =  /2.
/2
/2
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48. Critical z Values for Two-Tailed Test:  = 0.05
H0:  = k
H1:   k
If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis –
The critical regions: z < – 1.96 with z > 1.96.
143

49. Critical z Values for Two-Tailed Test:  = 0.01
H0:  = k
H1:   k
If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis –
The critical regions: z < – 2.58 with z > 2.58.

50. Critical z Value for Right-Tailed Test:  = 0.05
H0:  = k
H1:  > k
If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis
The critical region: z >
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51. Critical z Value for Right-Tailed Test:  = 0.01
H0:  = k
H1:  > k
If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis
The critical region: z > 2.33.

52. Critical z Value for Left-Tailed Test:  = 0.05
H0:  = k
H1:  < k
If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis
The critical region: z < – 1.645
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53. Critical z Value for Left-Tailed Test:  = 0.01
H0:  = k
H1:  < k
If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis
The critical region: z < – 2.33

54. Hypothesis Test Example
Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = 0.05.
A random sample of 49 students has a mean age of 26 years with a standard deviation of 2.3 years.

55. Hypothesis Test Example
Test H0:  = 28
Against H1:   28
Perform a ________-tailed test.
two

56. Hypothesis Test Example
Test H0:  = 28
Against H1:   28
Perform a ________-tailed test.
two
Using  = 0.05
Critical z value(s) = _________
±1.96
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57. Critical z Values for Two-Tailed Test:  = 0.05
H0:  = 28
H1:   28
If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis –
The critical regions: z < – 1.96 with z > 1.96.

58. Since z < – 1.96, we _________ the null hypothesis.
Sample Results
reject
Since z < – 1.96, we _________ the null hypothesis.
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59. Hypothesis Test Example
Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = 0.05.
A random sample of 49 students has a mean age of years with a standard deviation of 2.3 years.

60. Hypothesis Test Example
Test H0:  = 28
Against H1:   28
So, perform a two-tailed test.
Using  = Critical z values = 1.96
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61. Sample Results
Since the test statistic is neither < – 1.96 nor > 1.96 , we _______________ the null hypothesis.
do not reject
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62. Hypothesis Test Example
The manufacturer of light bulbs claims that they will burn for 1000 hours. I will test a sample of the bulbs before deciding whether to keep them. The bulbs will be returned to the manufacturer only if my sample indicates that they will burn less than 1000 hours.

63. Hypothesis Test Example
The manufacturer of light bulbs claims that they will burn for 1000 hours. ...The bulbs will be returned ... if my sample indicates that they will burn less than 1000 hours.
H0:  = 1000
H1:  < 1000

64. Hypothesis Test Example
Test H0:  = 1000
Against H1:  < 1000
Perform a ____-tailed test.)
left
–2.33
Using  = (So, critical z value = _______)
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65. Critical z Value for Left-Tailed Test:  = 0.01
H0:  = 1000
H1:  < 1000
If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis
The critical region: z < – 2.33

66. Sample Results
Since the test statistic is not < – 2.33 we _____________ the null hypothesis.
do not reject
162

67. Comparison of Critical z Values for Left-Tailed Tests:
 = 0.01 and  = 0.05 – –
.01
.05

68. In our last hypothesis test example, we calculated z = – 1.76.
Since we were using  = 0.01, the boundary of the critical region was
– 2.33.
Our conclusion was not to reject the null hypothesis.
Had we been using  = 0.05, our conclusion would have been to reject H0.

69. Comparison of Critical z Values for Left-Tailed Tests:
 = 0.01 and  = 0.05 – –
 =.01
 = .05
z = – 1.76
Reject H0.
z = – 1.76
Do not reject H0.

70. Statistical Significance
If we reject H0, we say that the data collected in the hypothesis testing process are statistically significant.
If we do not reject H0, we say that the data collected in the hypothesis testing process are not statistically significant.