1. STATISTICS INFORMED DECISIONS USING DATA
Fifth Edition
Chapter 10
Hypothesis Tests Regarding a Parameter
Copyright © 2018, 2014, 2011 Pearson Education, Inc. All Rights Reserved

2. 10.1 The Language of Hypothesis Testing Learning Objectives
1. Determine the null and alternative hypotheses 2. Explain Type I and Type II errors 3. State conclusions to hypothesis tests

3. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (1 of 17)
A hypothesis is a statement regarding a characteristic of one or more populations. In this chapter, we look at hypotheses regarding a single population parameter.

4. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (2 of 17)
Examples of Claims Regarding a Characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
Source: ReadersDigest.com poll created on 2008/05/02

5. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (3 of 17)
Examples of Claims Regarding a Characteristic of a Single Population
According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

6. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (4 of 17)
Examples of Claims Regarding a Characteristic of a Single Population
Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

7. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (5 of 17)
CAUTION!
If population data are available, there is no need for inferential statistics.

8. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (6 of 17)
Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

9. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (7 of 17)
Steps in Hypothesis Testing
Make a statement regarding the nature of the population.
Collect evidence (sample data) to test the statement.
Analyze the data to assess the plausibility of the statement.

10. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (8 of 17)
The null hypothesis, denoted H0, is a statement to be tested. The null hypothesis is a statement of no change, no effect, or no difference and is assumed true until evidence indicates otherwise.

11. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (9 of 17)
The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support.

12. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (10 of 17)
In this chapter, there are three ways to set up the null and alternative hypotheses:
Equal versus not equal hypothesis (two-tailed test)
H0: parameter = some value
H1: parameter ≠ some value

13. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (11 of 17)
In this chapter, there are three ways to set up the null and alternative hypotheses:
Equal versus less than (left-tailed test)
H0: parameter = some value
H1: parameter < some value

14. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (12 of 17)
In this chapter, there are three ways to set up the null and alternative hypotheses:
Equal versus greater than (right-tailed test)
H0: parameter = some value
H1: parameter > some value

15. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (13 of 17)
“In Other Words”
The null hypothesis is a statement of status quo or no difference and always contains a statement of equality. The null hypothesis is assumed to be true until we have evidence to the contrary. We seek evidence that supports the statement in the alternative hypothesis.

16. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (14 of 17)
Parallel Example 2: Forming Hypotheses
For each of the following claims, determine the null and alternative hypotheses. State whether the test is two-tailed, left-tailed or right-tailed.
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.
Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

17. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (15 of 17)
Solution
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
The hypothesis deals with a population proportion, p. If the percentage participating in charity work is no different than in 2008, it will be 0.62 so the null hypothesis is H0: p = 0.62.
Since the researcher believes that the percentage is different today, the alternative hypothesis is a two-tailed hypothesis: H1: p ≠ 0.62.

18. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (16 of 17)
Solution
b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.
The hypothesis deals with a population mean, μ. If the mean call length on a cellular phone is no different than in 2006, it will be 3.25 minutes so the null hypothesis is H0: μ = 3.25.
Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H1: μ > 3.25, a right-tailed test.

19. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Determine the Null and Alternative Hypotheses (17 of 17)
Solution
c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.
The hypothesis deals with a population standard deviation, σ. If the standard deviation with the new equipment has not changed, it will be 0.23 ounces so the null hypothesis is H0: σ = 0.23.
Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H1: σ < 0.23, a left-tailed test.

20. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (1 of 10)
Four Outcomes from Hypothesis Testing
Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct.

21. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (2 of 10)
Four Outcomes from Hypothesis Testing
Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct.

22. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (3 of 10)
Four Outcomes from Hypothesis Testing
Reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error.

23. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (4 of 10)
Four Outcomes from Hypothesis Testing
Do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a Type II error.

24. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (5 of 10)
Parallel Example 3: Type I and Type II Errors
For each of the following claims, explain what it would mean to make a Type I error. What would it mean to make a Type II error?
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

25. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (6 of 10)
Solution
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.
A Type I error is made if the researcher concludes that p ≠ 0.62 when the true proportion of Americans 18 years or older who participated in some form of charity work is currently 62%.
A Type II error is made if the sample evidence leads the researcher to believe that the current percentage of Americans 18 years or older who participated in some form of charity work is still 62% when, in fact, this percentage differs from 62%.

26. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (7 of 10)
Solution
According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was minutes. A researcher believes that the mean length of a call has increased since then.
A Type I error occurs if the sample evidence leads the researcher to conclude that μ > 3.25 when, in fact, the actual mean call length on a cellular phone is still 3.25 minutes.
A Type II error occurs if the researcher fails to reject the hypothesis that the mean length of a phone call on a cellular phone is 3.25 minutes when, in fact, it is longer than 3.25 minutes.

27. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (8 of 10)
α = P(Type I Error) = P(rejecting H0 when H0 is true) β = P(Type II Error) = P(not rejecting H0 when H1 is true)

28. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (9 of 10)
The probability of making a Type I error, α, is chosen by the researcher before the sample data is collected. The level of significance, α, is the probability of making a Type I error.

29. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing Explain Type I and Type II Errors (10 of 10)
“In Other Words”
As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.

30. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing State Conclusions to Hypothesis Tests (1 of 4)
CAUTION!
We never “accept” the null hypothesis, because, without having access to the entire population, we don’t know the exact value of the parameter stated in the null. Rather, we say that we do not reject the null hypothesis. This is just like the court system. We never declare a defendant “innocent”, but rather say the defendant is “not guilty”.

31. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing State Conclusions to Hypothesis Tests (2 of 4)
Parallel Example 4: Stating the Conclusion
According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.
Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion.
Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.

32. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing State Conclusions to Hypothesis Tests (3 of 4)
Solution
Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion.
The statement in the alternative hypothesis is that the mean call length is greater than 3.25 minutes. Since the null hypothesis (μ = 3.25) is rejected, there is sufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes.

33. 10. 1 The Language of Hypothesis Testing 10. 1
10.1 The Language of Hypothesis Testing State Conclusions to Hypothesis Tests (4 of 4)
Solution
b) Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.
Since the null hypothesis (μ = 3.25) is not rejected, there is insufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes. In other words, the sample evidence is consistent with the mean call length equaling 3.25 minutes.

34. 10.2 Hypothesis Tests for a Population Proportion Learning Objectives
1. Explain the logic of hypothesis testing 2. Test the hypotheses about a population proportion 3. Test hypotheses about a population proportion using the binomial probability distribution.

35. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (1 of 15)

36. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (2 of 15)
When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

37. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (3 of 15)
To determine if a sample proportion of is statistically significant, we build a probability model.

38. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (4 of 15)
Sampling distribution of the sample proportion

39. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (5 of 15)
The Logic of the Classical Approach
We may consider the sample evidence to be statistically significant (or sufficient) if the sample proportion is too many standard deviations, say 2, above the assumed population proportion of 0.5.

40. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (6 of 15)
Recall that our simple random sample yielded a sample proportion of 0.534, so

41. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (7 of 15)
Why does it make sense to reject the null hypothesis if the sample proportion is more than 2 standard deviations away from the hypothesized proportion? The area under the standard normal curve to the right of z = 2 is

42. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (8 of 15)
If the null hypothesis were true (population proportion is 0.5), then 1 − = = 97.72% of all sample proportions will be less than (0.016) = and only 2.28% of the sample proportions will be more than

43. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (9 of 15)
Hypothesis Testing Using the Classical Approach
If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis.

44. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (10 of 15)
Definition
A P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true.

45. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (11 of 15)
Hypothesis Testing Using the P-Value Approach
If the probability of getting a sample statistic as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis.

46. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (12 of 15)
The Logic of the P-Value Approach
A second criterion we may use for testing hypotheses is to determine how likely it is to obtain a sample proportion of or higher from a population whose proportion is 0.5. If a sample proportion of or higher is unlikely (or unusual), we have evidence against the statement in the null hypothesis. Otherwise, we do not have sufficient evidence against the statement in the null hypothesis.

47. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (13 of 15)
We can compute the probability of obtaining a sample proportion of or higher from a population whose proportion is 0.5 using the normal model.

48. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (14 of 15)
The value is called the P-value, which means about 2 samples in 100 will give a sample proportion as high or higher than the one we obtained if the population proportion really is 0.5. Because these results are unusual, we take this as evidence against the statement in the null hypothesis.

49. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Explain the Logic of Hypothesis Testing (15 of 15)
Hypothesis Testing Using the P-value Approach
If the probability of getting a sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis.

50. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (1 of 24)
Recall:
The best point estimate of p, the proportion of the population with a certain characteristic, is given by

51. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (2 of 24)
provided that the following requirements are satisfied:
The sample is a simple random sample.
​np(1−p) ≥ 10.
The sampled values are independent of each other.

52. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (3 of 24)
Testing Hypotheses Regarding a Population Proportion, p
To test hypotheses regarding the population proportion, we can use the steps that follow, provided that:
the sample is obtained by simple random sampling.
np0(1 − p0) ≥ 10.
the sampled values are independent of each other.

53. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (4 of 24)
Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:
Two-Tailed
Left-Tailed
Right-Tailed
H0: p = p0
H1: p ≠ p0
H1: p < p0
H1: p > p0
Note: p0 is the assumed value of the population proportion.

54. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (5 of 24)
Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error.

55. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (6 of 24)
Classical Approach
Step 3: Compute the test statistic

56. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (7 of 24)
Classical Approach
Use Table V to determine the critical value.
Two-Tailed

57. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (8 of 24)
Classical Approach
Left-Tailed

58. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (9 of 24)
Classical Approach
Right-Tailed

59. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (10 of 24)
Classical Approach
Step 4: Compare the critical value with the test statistic:

60. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (11 of 24)
P-Value Approach
By Hand Step 3: Compute the test statistic.

61. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (12 of 24)
P-Value Approach
Use Table V to estimate the P-value.
Two-Tailed

62. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (13 of 24)
P-Value Approach
Left-Tailed

63. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (14 of 24)
P-Value Approach
Right-Tailed

64. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (15 of 24)
P-Value Approach
Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value.
The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, MINITAB, Excel, and StatCrunch are in the Technology Step-by-Step in the text.

65. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (16 of 24)
P-Value Approach
Step 4: If the P-value < α, reject the null hypothesis.

66. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (17 of 24)
P-Value Approach
Step 5: State the conclusion.

67. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (18 of 24)
Parallel Example 1: Testing a Hypothesis about a Population Proportion: Large Sample Size
In 1997, 46% of Americans said they did not trust the media “when it comes to reporting the news fully, accurately and fairly”. In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. At the α = 0.05 level of significance, is there evidence to support the claim that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997?
Source: Gallup Poll

68. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (19 of 24)
Solution
We want to know if p > First, we must verify the requirements to perform the hypothesis test:
This is a simple random sample.
​np0(1 − p0) = 1010(0.46)(1 − 0.46) = > 10
Since the sample size is less than 5% of the population size, the assumption of independence is met.

69. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (20 of 24)
Solution
Step 1: H0: p = 0.46 versus H1: p > 0.46
Step 2: The level of significance is α = 0.05.

70. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (21 of 24)
Solution: Classical Approach
Since this is a right-tailed test, we determine the critical value at the α = 0.05 level of significance to be z0.05 =
Step 4: Since the test statistic, z0 = 3.83, is greater than the critical value 1.645, we reject the null hypothesis.

71. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (22 of 24)
Solution: P-Value Approach
Step 3: Since this is a right-tailed test, the P-value is the area under the standard normal distribution to the right of the test statistic z0 = That is, P-value = P(Z > 3.83) ≈ 0.
Step 4: Since the P-value is less than the level of significance, we reject the null hypothesis.

72. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (23 of 24)
Solution
Step 5: There is sufficient evidence at the α = 0.05 level of significance to conclude that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997.

73. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test Hypotheses about a Population Proportion (24 of 24)
Two-Tailed Hypothesis Testing Using Confidence Intervals When testing H0: p = p0 versus H1: p ≠ p0 if a (1 − α) • 100% confidence interval contains p0, do not reject the null hypothesis. However, if the confidence interval does not contain p0, conclude that p ≠ p0 at the level of significance, α.

74. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (1 of 7)
We stated that an event was unusual if the probability of observing the event was less than This criterion is based on the P-value approach to testing hypotheses; the probability that we computed was the P-value. We use this same approach to test hypotheses regarding a population proportion for small samples.

75. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (2 of 7)
Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size
In 2006, 10.5% of all live births in the United States were to mothers under 20 years of age. A sociologist claims that births to mothers under 20 years of age is decreasing. She conducts a simple random sample of 34 births and finds that 3 of them were to mothers under 20 years of age. Test the sociologist’s claim at the α = 0.01 level of significance.

76. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (3 of 7)
Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size
Approach:
Step 1: Determine the null and alternative hypotheses

77. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (4 of 7)
Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size
Approach:
Step 3: Compute the P-value. For right-tailed tests, the P-value is the probability of obtaining x or more successes. For left-tailed tests, the P-value is the probability of obtaining x or fewer successes. The P-value is always computed with the proportion given in the null hypothesis.
Step 4: If the P-value is less than the level of significance, α, we reject the null hypothesis.

78. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (5 of 7)
Solution
Step 1: H0: p = versus H1: p < 0.105

79. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (6 of 7)
Solution

80. 10. 2 Hypothesis Tests for a Population Proportion 10. 2
10.2 Hypothesis Tests for a Population Proportion Test hypotheses about a population proportion using the binomial probability distribution (7 of 7)
Solution
Step 4: The P-value = 0.51 is greater than the level of significance so we do not reject H0. There is insufficient evidence to conclude that the percentage of live births in the United States to mothers under the age of 20 has decreased below the 2006 level of 10.5%.

81. 10.3 Hypothesis Tests for a Population Mean Learning Objectives
1. Test hypotheses about a mean 2. Understand the difference between statistical significance and practical significance.

82. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (1 of 39)
To test hypotheses regarding the population mean assuming the population standard deviation is unknown, we use the t-distribution rather than the Z-distribution. When we replace σ with s,

83. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (2 of 39)
Properties of the t-Distribution
The t-distribution is different for different degrees of freedom.

84. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (3 of 39)
Properties of the t-Distribution
The t-distribution is centered at 0 and is symmetric about 0.

85. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (4 of 39)
Properties of the t-Distribution

86. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (5 of 39)
Properties of the t-Distribution
As t increases (or decreases) without bound, the graph approaches, but never equals, 0.

87. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (6 of 39)
Properties of the t-Distribution
The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution because using s as an estimate of σ introduces more variability to the t-statistic.

88. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (7 of 39)
Properties of the t-Distribution
As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s get closer to the values of σ by the Law of Large Numbers.

89. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (8 of 39)
Testing Hypotheses Regarding a Population Mean
To test hypotheses regarding the population mean, we use the following steps, provided that:
The sample is obtained using simple random sampling.
The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).
The sampled values are independent of each other.

90. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (9 of 39)
Step 1: Determine the null and alternative hypotheses. Again, the hypotheses can be structured in one of three ways:
Two-Tailed
Left-Tailed
Right-Tailed
H0: µ = µ0
H1: µ ≠ µ0
H1: µ < µ0
H1: µ > µ0
Note: µ0 is the assumed value of the population mean.

91. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (10 of 39)
Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error.

92. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (11 of 39)
Classical Approach
Step 3: Compute the test statistic

93. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (12 of 39)
Classical Approach
Use Table VII to determine the critical value.

94. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (13 of 39)
Classical Approach
Two-Tailed

95. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (14 of 39)
Classical Approach
Left-Tailed

96. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (15 of 39)
Classical Approach
Right-Tailed

97. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (16 of 39)
Classical Approach
Step 4: Compare the critical value with the test statistic:

98. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (17 of 39)
P-Value Approach
By Hand Step 3: Compute the test statistic

99. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (18 of 39)
P-Value Approach
Use Table VII to approximate the P-value.

100. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (19 of 39)
P-Value Approach Two-Tailed

101. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (20 of 39)
P-Value Approach Left-Tailed

102. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (21 of 39)
P-Value Approach Right-Tailed

103. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (22 of 39)
P-Value Approach
Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, MINITAB, Excel, and StatCrunch, are in the Technology Step-by- Step in the text.

104. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (23 of 39)
P-Value Approach
Step 4: If the P-value < α, reject the null hypothesis.

105. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (24 of 39)
P-Value Approach
Step 5: State the conclusion.

106. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (25 of 39)
The procedure is robust, which means that minor departures from normality will not adversely affect the results of the test. However, for small samples, if the data have outliers, the procedure should not be used.

107. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (26 of 39)
Parallel Example 1: Testing a Hypothesis about a Population Mean, Large Sample
Assume the resting metabolic rate (RMR) of healthy males in complete silence is 5710 kJ/day. Researchers measured the RMR of 45 healthy males who were listening to calm classical music and found their mean RMR to be with a standard deviation of
At the α = 0.05 level of significance, is there evidence to conclude that the mean RMR of males listening to calm classical music is different than 5710 kJ/day?

108. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (27 of 39)
Solution
We assume that the RMR of healthy males is 5710 kJ/day. This is a two-tailed test since we are interested in determining whether the RMR differs from 5710 kJ/day.
Since the sample size is large, we follow the steps for testing hypotheses about a population mean for large samples.

109. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (28 of 39)
Solution
Step 1: H0: μ = 5710 versus H1: μ ≠ 5710
Step 2: The level of significance is α = 0.05.

110. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (29 of 39)
Solution: Classical Approach
Since this is a two-tailed test, we determine the critical values at the α = 0.05 level of significance with n − 1 = 45 − 1 = 44 degrees of freedom to be approximately −t0.025 = −2.021 and t0.025 =
Step 4: Since the test statistic, t0 = −0.013, is between the critical values, we fail to reject the null hypothesis.

111. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (30 of 39)
Solution: P-value Approach
Step 3: Since this is a two-tailed test, the P-value is the area under the t-distribution with n − 1 = 44 degrees of freedom to the left of −t0.025 = −0.013 and to the right of t0.025 =
That is, P-value = P(t < −0.013) + P(t > 0.013)
= 2 P(t > 0.013) < P-value.
Step 4: Since the P-value is greater than the level of significance (0.05 < 0.5), we fail to reject the null hypothesis.

112. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (31 of 39)
Solution
Step 5: There is insufficient evidence at the α = 0.05 level of significance to conclude that the mean RMR of males listening to calm classical music differs from 5710 kJ/day.

113. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (32 of 39)
Parallel Example 3: Testing a Hypothesis about a Population Mean, Small Sample
According to the United States Mint, quarters weigh 5.67 grams. A researcher is interested in determining whether the “state” quarters have a weight that is different from 5.67 grams. He randomly selects 18 “state” quarters, weighs them and obtains the following data.
At the α = 0.05 level of significance, is there evidence to conclude that state quarters have a weight different than 5.67 grams?

114. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (33 of 39)
Solution
We assume that the weight of the state quarters is 5.67 grams. This is a two-tailed test since we are interested in determining whether the weight differs from 5.67 grams.
Since the sample size is small, we must verify that the data come from a population that is normally distributed with no outliers before proceeding to Steps 1 to 5.

115. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (34 of 39)
Assumption of normality appears reasonable.

116. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (35 of 39)
No outliers.

117. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (36 of 39)
Solution
Step 1: H0: μ = 5.67 versus H1: μ ≠ 5.67
Step 2: The level of significance is α = 0.05.
Step 3: From the data, the sample mean is calculated to be and the sample standard deviation is s = The test statistic is

118. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (37 of 39)
Solution: Classical Approach
Step 3: Since this is a two-tailed test, we determine the critical values at the α = 0.05 level of significance with n − 1 = 18 − 1 = 17 degrees of freedom to be −t0.025 = −2.11 and t0.025 = 2.11.
Step 4: Since the test statistic, t0 = 2.75, is greater than the critical value 2.11, we reject the null hypothesis.

119. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (38 of 39)
Solution: P-Value Approach
Step 3: Since this is a two-tailed test, the P-value is the area under the t-distribution with n − 1 = 17 degrees of freedom to the left of −t0.025 = −2.75 and to the right of t0.025 = That is,
P-value = P(t < −2.75) + P(t > 2.75)
= 2 P(t > 2.75) < P-value < 0.02.
Step 4: Since the P-value is less than the level of significance (0.02 < 0.05), we reject the null hypothesis.

120. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Test Hypotheses about a Mean (39 of 39)
Solution
Step 5: There is sufficient evidence at the α = 0.05 level of significance to conclude that the mean weight of the state quarters differs from 5.67 grams.

121. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (1 of 7)
When a large sample size is used in a hypothesis test, the results could be statistically significant even though the difference between the sample statistic and mean stated in the null hypothesis may have no practical significance.

122. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (2 of 7)
Practical significance refers to the idea that, while small differences between the statistic and parameter stated in the null hypothesis are statistically significant, the difference may not be large enough to cause concern or be considered important.

123. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (3 of 7)
Parallel Example 7: Statistical versus Practical Significance In 2003, the average age of a mother at the time of her first childbirth was To determine if the average age has increased, a random sample of 1200 mothers is taken and is found to have a sample mean age of 25.5 with a standard deviation of 4.8, determine whether the mean age has increased using a significance level of α = 0.05.

124. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (4 of 7)
Solution
Step 1: To determine whether the mean age has increased, this is a right-tailed test with
H0: μ = 25.2 versus H1: μ > 25.2.
Step 2: The level of significance is α = 0.05.
Step 3: Recall that the sample mean is The test statistic is then

125. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (5 of 7)
Solution
Step 3: Since this is a right-tailed test, the critical value with α = 0.05 and 1200 − 1 = 1199 degrees of freedom is
P-value = P(t > 2.17).
From Table VI: .01 < P-value < .025
Step 4: Because the P-value is less than the level of significance, 0.05, we reject the null hypothesis.

126. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (6 of 7)
Solution
Step 5: There is sufficient evidence at the 0.05 significance level to conclude that the mean age of a mother at the time of her first childbirth is greater than 25.2.
Although we found the difference in age to be significant, there is really no practical significance in the age difference (25.2 versus 25.5).

127. 10. 3 Hypothesis Tests for a Population Mean 10. 3
10.3 Hypothesis Tests for a Population Mean Understand the difference between Statistical Significance and Practical Significance (7 of 7)
Large Sample Sizes
Large sample sizes can lead to statistically significant results while the difference between the statistic and parameter is not enough to be considered practically significant.

128. 10.4 Putting It Together: Which Method Do I Use? Learning Objective
1. Determine the appropriate hypothesis test to perform

129. 10. 4 Putting It Together: Which Method Do I Use. 10. 4
10.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform