SLIDES PREPARED

1
STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

2
Chapter 12

• Hypothesis Tests Large Samples

3
Outline

• Do I Need to Read This Chapter? You should read
  the Chapter if you would like to learn about
• 12-1 Some Terms Associated with
• Hypothesis Testing.
• 12-2 Large-Sample Hypothesis Tests
• for a Single Population Proportion.
• 12-3 Large-Sample Hypothesis Tests
• for a Single Population Mean.

4
Outline

• Do I Need to Read This Chapter? You should read
  the Chapter if you would like to learn about
• 12-4 Large-Sample Hypothesis Tests
• for the Difference between Two
• Population Proportions.
• 12-4 Large-Sample Hypothesis Tests
• for the Difference between Two
• Population Means.

5
Objectives

• To perform hypothesis tests for a single
  population proportion and a single population
  mean.
• To perform hypothesis tests for the difference
  between two population proportions and the
  difference between two population means.

6
12-1 Some Terms Associated with
Hypothesis Testing

• Every situation that requires a hypothesis test
  starts with a statement of a hypothesis.
• Explanation of the term statistical hypothesis
  A statistical hypothesis is an opinion about a
  population parameter.

7
12-1 Some Terms Associated with
Hypothesis Testing

• There are two types of hypotheses
• Null Hypothesis, denoted by H0.
• Alternative Hypothesis, denoted by H1.
• Explanation of the term null hypothesis The
  null hypothesis states that there is no
  difference between a parameter (or parameters)
  and a specific value.

8
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term alternative hypothesis
  The alternative hypothesis states that there is a
  precise difference between a parameter (or
  parameters) and a specific value.
• Note During a hypothesis test, the hypothesis
  that is assumed to be true is the null
  hypothesis.

9
12-1 Some Terms Associated with
Hypothesis Testing

• After the hypotheses are stated, the next step is
  to design the study.
• An appropriate statistical test will be selected.
• The level of significance will be chosen.
• And a plan to conduct the study will be
  formulated.
• To make an inference for the study, the
  statistical test and level of significance are
  used.

10
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term statistical test A
  statistical test uses the data collected from the
  study to make a decision about the null
  hypothesis.
• Note This decision will be to reject or not to
  reject the null hypothesis.
• We will need to to compute a test value or test
  statistic in order to make the decision.
• The formula that is used to compute this value
  will vary depending on the statistical test.

11
12-1 Some Terms Associated with
Hypothesis Testing

• POSSIBLE OUTCOMES FOR A HYPOTHESIS TEST

12
12-1 Some Terms Associated with
Hypothesis Testing
TYPES OF ERROR FOR A HYPOTHESIS TEST

• Observe from the table that there are two ways of
  making a mistake when doing a hypothesis test
  Type I and Type II errors.
• Explanation of the term Type I error A Type I
  error occurs if a true null hypothesis is
  rejected.

13
12-1 Some Terms Associated with
Hypothesis Testing
TYPES OF ERROR FOR A HYPOTHESIS TEST

• Explanation of the term Type II error A Type
  II error occurs if a false null hypothesis is not
  rejected.

14
12-1 Some Terms Associated with
Hypothesis Testing

• When we reject or do not reject the null
  hypothesis, how confident are we that we are
  making the correct decision?
• This question can be answered by the specified
  level of significance.

15
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term Level of Significance
  The level of significance, denoted by ?, is the
  probability of a Type I error.
• Typical values for ? are 0.1, 0.05, and 0.01.
• For example, if ? 0.1 for a test, and the null
  hypothesis is rejected, then one would be 90
  confident that this is the correct decision.

16
12-1 Some Terms Associated with
Hypothesis Testing

• Once the level of significance is selected, a
  critical value for the appropriate test is
  selected from a table (or maybe generated by a
  statistical software or calculator dedicated to
  statistics like the TI-83).
• Explanation of the term critical value A
  critical value separates the critical region from
  the non-critical region.

17
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term critical or rejection
  region A critical or rejection region is a range
  of test statistic values for which the null
  hypothesis will be rejected.
• This range of values will indicate that there is
  a significant or large enough difference between
  the postulated parameter value and the
  corresponding point estimate for the parameter.

18
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term non-critical or
  non-rejection region A non-critical or
  non-rejection region is a range of test statistic
  values for which the null hypothesis will not be
  rejected.
• This range of values will indicate that there is
  not a significant or large enough difference
  between the postulated parameter value and the
  corresponding point estimate for the parameter.

19

• GRAPHICAL DISPLAY OF A CRITICAL AND NON-CRITICAL
  REGION

20
12-1 Some Terms Associated with
Hypothesis Testing

• There are two broad classifications of hypothesis
  tests (a) one-tailed tests, and (b) two-tailed
  tests.
• Explanation of the term one-tailed test A
  one-tailed test points out that the null
  hypothesis should be rejected when the test
  statistic value is in the critical region on one
  side of the parameter value being tested.

21
12-1 Some Terms Associated with
Hypothesis Testing

• Explanation of the term two-tailed test A
  two-tailed test points out that the null
  hypothesis should be rejected when the test
  statistic value is in either of the two critical
  regions.

22
Five-Step Procedure for Hypothesis Testing

• Step 1 State the null hypothesis H0.
• Step 2 State the alternative hypothesis H1.
• Step 3 State the formula for the test statistic
  (T.S) and compute its value.
• Step 4 State the decision rule (D.R) for
  rejecting the null hypothesis for a given level
  of significance.
• Step 5 State a conclusion in the context of the
  information given in the problem.

23
12-2 Large Sample Test for a
Population Proportion

• Here we will present tests of hypotheses that
  will enable us to determine, based on sample
  data, whether the true value of a population
  proportion equals a given constant.
• Samples of size n will be obtained, and the
  number or proportion of successes observed.

24
12-2 Large Sample Test for a
Population Proportion

• It will be assumed that the trials are
  independent and that the probability of success
  is the same for each trial.
• That is, we are assuming that we have a binomial
  experiment, and we are testing hypotheses about
  the parameter p of a binomial population.

25
12-2 Some z-values for different values of ?

• The following table lists values for z? and z?/2
  when ? 0.1, 0.05, and 0.01.
• These values can be obtained from the standard
  normal tables.

26
12-2 Large Sample Test for a
Population Proportion Experimental Display
p population proportion
Population
Sample
n sample size x number of successes
27
12-2 One-tailed (right-tailed) Test for a
Population Proportion - Summary

• H0 p ? p0 (where p0 is a specified proportion
  value)
• H1 p gt p0
• T.S z (x np0)/?np0(1 p0)
• D.R For a specified significance level ?, reject
  the null hypothesis if the computed test
  statistic value z gt z?.
• Conclusion

28
12-2 One-tailed (right-tailed) Test for a
Population Proportion - Example

• Your teacher claims that 60 of American males
  are married. You fell that the proportion is
  higher. In a random sample of 100 American
  males, 65 of them were married. Test your
  teachers claim at the 5 level of significance.

29
12-2 One-tailed (right-tailed) Test for a
Population Proportion - Example

• Summary n 100, x (number of successes) 65, ?
  0.05, z? 1.645, and p0 0.6.
  Also, ?np0(1 p0) 4.8990.
• Since you would like to establish that the
  proportion is higher, the alternative hypothesis
  should reflect this belief.

30
12-2 One-tailed (right-tailed) Test for a
Population Proportion - Example

• H0 p ? 0.6
• H1 p gt 0.6
• T.S z (x np0)/?np0(1 p0) (65 -
  100?0.6)/4.8990 1.0206.
• D.R For a significance level of ? 0.05, reject
  the null hypothesis if the computed test
  statistic value z 1.0206 gt z0.05
  1.645.

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12-2 One-tailed (right-tailed) Test for a
Population Proportion - Example

• Conclusion Since 1.0206 lt 1.645, do not reject
  H0. There is insufficient sample evidence to
  refute your teachers claim. That is, there is
  insufficient sample evidence to claim that more
  than 60 of American males are married at the 5
  level of significance.
• Note Any difference between the sample
  proportion and the postulated proportion of 0.6
  may be due to chance.

32
12-2 One-tailed (right-tailed) Test for a
Population Proportion Display of the Rejection
(Critical) Region
33
12-2 One-tailed (left-tailed) Test for a
Population Proportion - Summary

• H0 p ? p0 (where p0 is a specified proportion
  value)
• H1 p lt p0
• T.S z (x np0)/?np0(1 p0)
• D.R For a specified significance level ?, reject
  the null hypothesis if the computed test
  statistic value z lt -z?.
• Conclusion

34
12-2 One-tailed (left-tailed) Test for a
Population Proportion - Example

• A preacher would like to establish that of people
  who pray, less than 80 pray for world peace. In
  a random sample of 110, 77 of them said that when
  they pray, they pray for world peace. Test at
  the 10 level of significance.

35
12-2 One-tailed (left-tailed) Test for a
Population Proportion - Example

• Summary n 110, x (number of successes) 77, ?
  0.10, z? 1.28, and p0 0.8. Also, ?np0(1
  p0) 4.1952.
• Since the preacher would like to establish that
  less than 80 of people pray for world peace, the
  alternative hypothesis should reflect this.

36
12-2 One-tailed (left-tailed) Test for a
Population Proportion - Example

• H0 p ? 0.8
• H1 p lt 0.8
• T.S z (x np0)/?np0(1 p0) (77 -
  110?0.8)/4.1952 -2.622.
• D.R For a significance level of ? 0.10, reject
  the null hypothesis if the computed test
  statistic value z -2.6220 lt -z0.1
  -1.28.

37
12-2 One-tailed (left-tailed) Test for a
Population Proportion - Example

• Conclusion Since 2.622 lt -1.28, reject H0.
  There is sufficient sample evidence to confirm
  your preachers claim. That is, there is
  sufficient sample evidence to claim that less
  than 80 of people, when they pray, pray for
  world peace at the 10 level of significance.
• Note This means that there is a significant
  difference between the sample proportion and the
  postulated proportion of 0.8.

38
12-2 One-tailed (left-tailed) Test for a
Population Proportion Display of the Rejection
(Critical) Region
39
12-2 Two-tailed Test for a Population Proportion
- Summary

• H0 p p0 (where p0 is a specified proportion
  value)
• H1 p ? p0
• T.S z (x np0)/?np0(1 p0)
• D.R For a specified significance level ?, reject
  the null hypothesis if the computed test
  statistic value z lt -z?/2 or if z gt
  z?/2.
• Conclusion

40
12-2 Two-tailed Test for a Population Proportion
- Example

• A researcher claims that 90 of people trust DNA
  testing. In a survey of 100 people, 91 of them
  said that they trusted DNA testing. Test the
  researchers claim at the 1 level of
  significance.

41
12-2 Two-tailed Test for a Population Proportion
- Example

• Summary n 100, x (number of successes) 91, ?
  0.01, z?/2 2.576, and p0 0.9.
  Also, ?np0(1 p0) 3.0.
• Since we are asked to test the researchers
  claim, this will be a two-tail test.

42
12-2 Two-tailed Test for a Population Proportion
- Example

• H0 p 0.9
• H1 p ? 0.9
• T.S z (x np0)/?np0(1 p0) (91 -
  100?0.9)/3 0.3333.
• D.R For a significance level of ? 0.01, reject
  the null hypothesis if the computed test
  statistic value z 0.3333 lt -z0.005
  -2.576 or if z 0.3333 gt z0.005 2.575.

43
12-2 Two-tailed Test for a Population Proportion
- Example

• Conclusion Since neither of the conditions is
  satisfied in the decision rule, do not reject H0.
  There is insufficient sample evidence to refute
  the researchers claim that the proportion of
  people who believe in DNA testing is equal to 90
  at the 1 level of significance.
• Note This means that there is not a significant
  difference between the sample proportion and the
  postulated proportion of 0.9.

44
12-2 Two-tailed Test for a Population Proportion
Display of the Rejection (Critical) Region
45
12-3 Large Sample Tests for a
Population Mean

• Here we will present tests of hypotheses that
  will enable us to determine, based on sample
  data, whether the true value of a population mean
  equals a given constant.
• Samples of size n will be obtained from a normal
  population.

46
12-3 Large Sample Tests for a
Population Mean

• It will be assumed that the sampling distribution
  for the sample means is approximately normally
  distributed.
• The tests require that the population standard
  deviation ? be known, but if ? is unknown, then n
  ? 30 (large sample), unless the sampling
  distribution is exactly normally distributed.

47
12-3 Large Sample Tests for a
Population Mean Experimental Display
? population mean

• population standard
• deviation (SD)

Sample
n sample size sample mean s sample SD.
Population
48
12-3 One-tailed (right-tailed) Test for a
Population Mean - Summary
Note This is a right- tail test because the
direction of the inequality in the alternative
hypothesis is to the right.
49
12-3 One-tailed (right-tailed) Test for a
Population Mean - Example

• Management of a large accounting firm claims that
  the mean salary of the firms accountants is
  greater than its competitors, which is 45,000.
  A random sample of 30 of the firms accountants
  yielded a mean salary of 46,500. Test the firms
  claim at the 5 level of significance. Assume
  that the standard deviation of the salaries for
  the firm is 5,200.

50
12-3 One-tailed (right-tailed) Test for a
Population Mean - Example

• Summary n 30, ? 5,200, ? 0.05,
  x-bar (sample mean) 46,500, z? 1.645, and ?0
  45,000. Also, ?/?n 949.3858.
• Since you would like to establish that the
  average salary for the firm is higher, the
  alternative hypothesis should reflect this
  belief.

51
12-3 One-tailed (right-tailed) Test for a
Population Mean - Example

• H0 ? ? 45,000
• H1 ? gt 45,000
• T.S z (x-bar ?)/?/?n (46,500
  45,000)/949.3858 1.58.
• D.R For a significance level of ? 0.05, reject
  the null hypothesis if the computed test
  statistic value z 1.58 gt z0.05
  1.645.

52
12-3 One-tailed (right-tailed) Test for a
Population mean - Example

• Conclusion Since 1.58 lt 1.645, do not reject H0.
  There is insufficient sample evidence to confirm
  your firms claim. That is, there is
  insufficient sample evidence to claim that the
  average salary for the firms accountants is
  greater than 45,000 at the 5 level of
  significance.
• Note Any difference between the sample mean and
  the postulated mean of 45,000 may be due to
  chance.

53
12-3 One-tailed (right-tailed) Test for a
Population Mean Display of the Rejection
(Critical) Region
54
12-3 One-tailed (Left-tailed) Test for a
Population Mean - Summary
Note This is a left- tail test because the
direction of the inequality in the alternative
hypothesis is to the left.
55
12-3 One-tailed (left-tailed) Test for a
Population Mean - Example

• A teachers union would like to establish that
  the average salary for high school teachers in a
  particular state is less than 32,500. A random
  sample of 100 public high school teachers in the
  particular state has a mean salary of 31,578.
  It is known from past history that the standard
  deviation of the salaries for the teachers in the
  state is 4,415. Test the unions claim at the
  5 level of significance.

56
12-3 One-tailed (left-tailed) Test for a
Population Mean - Example

• Summary n 100, ? 4,415, ? 0.05,
  x-bar (sample mean) 31,578, z? 1.645, and ?0
  32,500. Also, ?/?n 441.5.
• Since the union would like to establish that the
  average salary is less than 32,500, this will be
  a left-tailed test.

57
12-3 One-tailed (left-tailed) Test for a
Population Mean - Example

• H0 ? ? 32,500
• H1 ? lt 32,500
• T.S z (x-bar ?)/?/?n (31,578
  32,500)/441.5 -2.0883.
• D.R For a significance level of ? 0.05,
  reject the null hypothesis if the computed test
  statistic value z -2.0883 lt -z0.05
  -1.645.

58
12-3 One-tailed (left-tailed) Test for a
Population Mean - Example

• Conclusion Since 2.0883 lt -1.645, reject H0.
  There is sufficient sample evidence to support
  the claim that the average salary for high school
  teachers in the state is less than 32,500 at the
  5 level of significance.
• Note There is a significant difference between
  the sample mean and the postulated value of the
  population mean of 32,500.

59
12-3 One-tailed (left-tailed) Test for a
Population Mean Display of the Rejection
(Critical) Region
60
12-3 Two-tailed Test for a Population Mean -
Summary
Note This is a two- tail test because of the
not-equal symbol in the alternative hypothesis.
61
12-3 Two-tailed Test for a Population Mean -
Example

• The dean of students of a four-year college
  claims that the average distance commuting
  students travel to the campus is 32 miles. The
  commuting students feel otherwise. A sample of
  64 commuting students was randomly selected and
  yielded a mean of 35 miles and a standard
  deviation of 5 miles. Test the deans claim at
  the 5 level of significance.

62
12-3 Two-tailed Test for a Population Mean -
Example

• Summary n 64, s 5, ? 0.05, x-bar
  (sample mean) 35, z?/2 1.96, and ?0
  32. Also, s/?n 0.625.
• This will be a two-tailed test, since the
  students feel that the deans claim is not
  correct.
• Observe that whether the students feel that the
  average distance is less than 32 miles or more
  than 32 miles is not specified.

63
12-3 Two-tailed Test for a Population Mean -
Example

• H0 ? 32
• H1 ? ? 32
• T.S z (x-bar ?)/s/?n (35
  32)/0.625 4.8.
• D.R For a significance level of ? 0.05,
  reject the null hypothesis if the computed test
  statistic value z 4.8 lt -z0.025 -1.96 or if z
  4.8 gt z0.025 1.96 .

64
12-3 Two-tailed Test for a Population Mean -
Example

• Conclusion Since 4.8 gt 1.96, reject H0. There
  is sufficient sample evidence to refute the
  deans claim. The sample evidence supports the
  students claim that the average distance
  commuting students travel to the campus is not
  equal to 32 miles at the 5 level of
  significance.
• Note There is a significant difference between
  the sample mean and the postulated value of the
  population mean of 32 miles.

65
12-3 Two-tailed Test for a Population Mean
Display of the Rejection (Critical) Region
66
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions

• There may be problems in which one must decide
  whether the observed difference between two
  sample proportions is due to chance or whether
  the difference is due to the fact that the
  corresponding population proportions are not the
  same or that the proportions are from different
  populations. In this section we will discuss
  tests for such problems.

67
Point Estimate for the Difference Between Two
Population Proportions
p1
p2
n1 sample size x1 number of successes
n2 sample size x2 number of successes
Population 1
Population 2
68
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions

• NOTE Here, large sample is assumed when n1?p1 gt
  5, n1?(1 p1) gt 5, n2?p2 gt 5 and n2?(1 p2) gt
  5.

69
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions

• We can summarize the properties of the sampling
  distribution for the difference between two
  independent sample proportions with the following
  statements
• As the sample sizes n1 and n2 increase, the shape
  of the distribution of the differences of the
  sample proportions obtained from any population
  will approach a normal distribution.

70
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions

• The distribution of the differences of the sample
  proportions will have a mean and standard
  deviation given on the next slide.
• NOTE p1 and p2 are the respective population
  proportion of interest.

71
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions
72
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions

• These properties can aid us in testing for the
  difference between two population proportions for
  large samples.

73
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions
NOTE Depending on which test we are dealing
with, we will replace the sign with ? or
? in the null hypothesis.
74
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
Note In the null hypothesis we have replaced the
sign with ?.
75
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
- Example

• Example A study was conducted to determine
  whether remediation in basic mathematics enabled
  students to be more successful in an elementary
  statistics course. Success here means a student
  received a grade of C or better in the statistics
  course and remediation was for one-year prior to
  the statistics course.
• The next slide shows the results of the study.

76
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
- Example (continued).
77
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
- Example (continued).

• Test, at the 5 level of significance, whether
  remediation helped the students to be more
  successful.
• Let p1 be the proportion of students who were
  successful from the remediation group.

78
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
- Example (continued).

• Example (continued) Let p2 be the proportion of
  students who were successful from the
  non-remediation group.

79
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Right-tailed)
- Example (continued).
80
12-4 Display of the Rejection (Critical) Region
81
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
Note In the null hypothesis we have replaced the
sign with ?.
82
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
- Example

• Example A medical study was conducted to
  determine the effect of a cholesterol reducing
  medication when compared to a placebo. At the
  end of the study, the number of people who died
  of heart disease in each sample group was
  recorded. The next slide shows the results of the
  study.

83
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
- Example (continued).
84
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
- Example (continued).

• Test at the 1 significance level to determine
  whether the medication reduced the death rate.
• Let p1 be the proportion of patients who died in
  the medication group.

85
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
- Example (continued).

• Example (continued) Let p2 be the proportion of
  patients who died in the placebo group.

86
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Left-tailed)
- Example (continued).
87
12-4 Display of the Rejection (Critical) Region
88
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed)
89
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed) -
Example

• Example A researcher wants to determine whether
  there is a difference between the proportions of
  males and females who believe in outer-space
  aliens. The next slide shows the results of the
  study.

90
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed) -
Example (continued).
91
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed) -
Example (continued).

• Test at the 1 significance level.
• Let p1 be the proportion of males who believe in
  outer-space aliens.
• Let p2 be the proportion of females who believe
  in outer-space aliens.

92
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed) -
Example (continued).

• Example (continued)

93
12-4 Large-Sample Tests for the Difference
Between Two PopulationProportions (Two-tailed) -
Example (continued).
94
12-4 Display of the Rejection (Critical) Region
95
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans

• There may be problems in which one must decide
  whether the observed difference between two
  sample means is due to chance or whether the
  difference is due to the fact that the
  corresponding population means are not the same
  or that the means are from different populations.
  In this section we will discuss tests for such
  problems.

96
Point Estimate for the Difference Between Two
Population Means
?1
?2
n1 sample size x1- bar sample
mean
n2 sample size x2 bar sample
mean
Population 1
Population 2
97
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans

• NOTE Here, large sample is assumed when n1? 30
  and n2 ? 30 .

98
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans

• We can summarize the properties of the sampling
  distribution for the difference between two
  independent sample means with the following
  statements
• As the sample sizes n1 and n2 increase, the shape
  of the distribution of the differences of the
  sample means obtained from any population will
  approach a normal distribution.

99
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans

• The distribution of the differences of the sample
  means will have a mean and standard deviation
  given on the next slide.
• NOTE ?1 and ?2 are the respective population
  means of interest.

100
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans
101
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans

• These properties can aid us in testing for the
  difference between two population means for large
  samples.

102
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans
NOTE Depending on the test with which we are
dealing, we will replace the sign with ?
or ? in the null hypothesis.
103
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Right-tailed)
Note In the null hypothesis we have replaced the
sign with ?.
104
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Right-tailed) -
Example

• Example Two methods were used to teach a high
  school algebra course. A sample of 75 scores was
  selected for method 1, and a sample of 60 scores
  was selected for method 2.
• The next slide shows the results of the study.

105
12-5 Large-Sample Tests for the Difference
Between Two PopulationMean (Right-tailed) -
Example (continued).
106
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Right-tailed) -
Example (continued).

• Test, at the 1 level of significance, whether
  method 1 was more successful than method 2.

107
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Right-tailed) -
Example (continued).

• Example (continued)

108
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Right-tailed) -
Example (continued).
109
12-5 Display of the Rejection (Critical) Region
110
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Left-tailed)
Note In the null hypothesis we have replaced the
sign with ?.
111
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Two-tailed)
112
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Two-tailed) -
Example

• Example A random sample of size n1 36 selected
  from a normal distribution with standard
  deviation ?1 4 has a mean x1-bar 75. A
  second random sample of size n2 25 selected
  from a different normal distribution with
  standard deviation ?2 6 has a mean x2 bar
  85. Is there a significant difference between
  the population means at the 5 level of
  significance.

113
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Two-tailed) -
Example (continued).

• Example (continued)

114
12-5 Large-Sample Tests for the Difference
Between Two PopulationMeans (Two-tailed) -
Example (continued).
115
12-5 Display of the Rejection (Critical) Region
116
12-6 P-Value Approach to Hypothesis Testing

• With the advent of the computer and some
  calculators, certain probabilities can be readily
  computed and used to help make decisions in
  hypothesis tests.One such probability value is
  called the P-value.

117
12-6 P-Value Approach to Hypothesis Testing

• Explanation of the term P-value A P-value is
  the smallest significance level at which a null
  hypothesis may be rejected.

118
12-6 Determining P-Values

• For each of the above tests, we can compute a
  P-value.
• Right-tailed test P-value P(z gt z), where z
  is the computed value for the test statistic.
• Left-tailed test P-value P(z lt z), where z
  is the computed value for the test statistic.
• Two-tailed test P-value 2?P(z gt z), where
  z is the computed value for the test statistic.

119
12-6 Determining P-Values

• Example Compute the P-value for a right-tailed
  test with a test statistic value of z 1.0206.
• Solution The P-value P(z gt 1.0206) 0.5
  0.3461 0.1539.
• The area is shown on the next slide.

120
12-6 Determining P-Values
121
12-6 Determining P-Values

• Example Compute the P-value for a left-tailed
  test with a test statistic value of z -2.622.
• Solution The P-value P(z lt -2.622) 0.5
  0.4956 0.0044.
• The area is shown on the next slide.

122
12-6 Determining P-Values
123
12-6 Determining P-Values

• Example Compute the P-value for a two-tailed
  test with a test statistic value of z 0.3333.
• Solution The P-value 2?P(z gt 0.3333)
  2?(0.5 0.1293) 0.7414.
• The area is shown on the next slide.

124
12-6 Determining P-Values
125
12-6 Interpreting P-Values

• We can use the P-values to measure the strength
  of the evidence against the null hypothesis.
• The smaller the P-value, the stronger the
  evidence against the null hypothesis for us to
  reject it in favor of the alternative hypothesis.

126
12-6 Interpreting P-Values

• The following can be used to help make your
  decision when ? is the significance level.
• If P-value lt ? ? reject the null hypothesis.
• If P-value ? ? ? do not reject the null
  hypothesis.