Statistics for Business and Economics (14e)
Summary of Ch8, Ch9, and Ch10
• One Population:
Ch8. Interval Estimation – σ Known and Unknown
Ch9. Hypothesis Test – σ Known and Unknown
• Two Population:
Ch10. Interval Estimation & Hypothesis Test – σ Known and Unknown
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1
Statistics for Business and Economics (14e)
Summary of Interval Estimation Procedures for a Population Mean
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2
Statistics for Business and Economics (14e)
Summary of Forms for Null and Alternative Hypotheses
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population mean μ must take one
of the following three forms (where μ0 is the hypothesized value of the population
mean).
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3
Statistics for Business and Economics (14e)
Steps of Hypothesis Testing
Step 1. Develop the null and alternative hypotheses.
Step 2. Specify the level of significance α.
Step 3. Collect the sample data and compute the value of the test statistic.
p-Value Approach  When we can calculate z-score/-test statistics, find p-value (=area probability).
Step 4. Use the value of the test statistic to compute the p-value.
Step 5. Reject H0 if p-value ≤ α.
Critical Value Approach  when area probability (= p-value) is given, find the z-score value.
Step 4. Use the level of significance α to determine the critical value and the rejection rule.
Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0.
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4
Statistics for Business and Economics (14e)
One-Tailed Test : Step 5
Decision Rule 1: Reject 𝐻0 if p-value ≤ 𝜶.
Null Hypothesis: 𝐻0 : 𝜇 ≥ 3
Alternative Hypothesis: 𝐻𝑎 : 𝜇 < 3
 the p-value of .0038 resulted in the rejection of
the null hypothesis
 Reject 𝐻0 because p-value, 0.0038 < 0.01.
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5
Statistics for Business and Economics (14e)
One-Tailed Test : Step 5
Decision Rule 2: Reject 𝐻0 if 𝒛 ≤ 𝒛𝜶 .
From Step 3,
ҧ
𝑥−𝜇
z=
𝜎/ 𝑛
=
2.92−3
=-2.67
0.03
Reject H0 because z = −2.67 < zα = −2.33
𝒛𝟎.𝟎𝟏
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6
Statistics for Business and Economics (14e)
Two-Tailed Tests About a Population Mean: σ Known
Critical Value Approach
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7
Statistics for Business and Economics (14e)
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8
Statistics for Business and Economics (14e)
Estimating the Difference Between Two Population Means
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9
Statistics for Business and Economics (14e)
Interval Estimation of μ1 – μ2 when σ1 and σ2 are Known
Interval Estimate:
Point Estimator ± Margin Error
𝛼 is the level of significance.
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10
Statistics for Business and Economics (14e)
Hypothesis Tests About μ1 – μ2 when σ1 and σ2 are Known
A hypothesis test about the value of the difference in two population means 𝜇1 −𝜇2
must take one of the following three forms (where D0 is the hypothesized difference in
the population means).
Test Statistic:
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11
Statistics for Business and Economics (14e)
Interval Estimation of μ1 – μ2 when σ1 and σ2 are Unknown
Interval Estimate
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12
Statistics for Business and Economics (14e)
Hypothesis Tests About μ1 – μ2 when σ1 and σ2 are Unknown
A hypothesis test about the value of the difference in two population means 𝜇1 −𝜇2 must take one
of the following three forms (where D0 is the hypothesized difference in the population means).
Test Statistic:
with
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13
Statistics for Business and Economics (14e)
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14
Statistics for Business and Economics (14e)
Chapter 13 - ANalysis Of VAriance (ANOVA)
1 - Analysis of Variance and the Completely Randomized Design
2 - Multiple Comparison Procedures
• Fisher’s LSD
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15
Statistics for Business and Economics (14e)
An Introduction to Experimental Design and Analysis of Variance
• A factor is a variable that the experimenter has selected for investigation.
• A treatment is a level of a factor.
• Experimental units are the objects of interest in the experiment.
• A completely randomized design is an experimental design in which the treatments are
randomly assigned to the experimental units.
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16
Statistics for Business and Economics (14e)
--------
Population k
𝜇𝑘 , 𝑛𝑘 , 𝜎𝑘
𝑥ҧ𝑘 , 𝑠𝑘
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17
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• Analysis of Variance (ANOVA) can be used to test for the equality of three or
more population means.
• Data obtained from observational or experimental studies can be used for the
analysis.
• We want to use the sample results to test the following hypotheses:
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18
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• If 𝐻0 is rejected, we cannot conclude that all population means are different.
• Rejecting 𝐻0 means that at least two population means have different values.
• Assumptions for Analysis of Variance:
1. For each population, the response (dependent) variable is normally distributed.
2. The variance of the response variable, denoted σ2, is the same for all of the
populations.
3. The observations must be independent.
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19
Statistics for Business and Economics (14e)
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20
Statistics for Business and Economics (14e)
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21
Statistics for Business and Economics (14e)
ഥ = 60
𝒙
(ഥ
𝒙)
(𝒔𝟐 )
(s)
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22
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• Analysis of Variance (ANOVA) can be used to test for the equality of three or more
population means.
• Data obtained from observational or experimental studies can be used for the analysis.
• We want to use the sample results to test the following hypotheses:
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password-protected website or school-approved learning management system for classroom use.
23
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
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24
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• If 𝐻0 is rejected, we canNOT conclude that all population means are different.
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password-protected website or school-approved learning management system for classroom use.
25
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• If 𝐻0 is rejected, we canNOT conclude that all population means are different.
• Rejecting 𝐻0 means that at least two population means have different values.
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26
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
• If 𝐻0 is rejected, we canNOT conclude that all population means are different.
• Rejecting 𝐻0 means that at least two population means have different values.
• Assumptions for Analysis of Variance (i.i.d. = independent and identically distributed)
1. For each population, the response (dependent) variable is normally distributed.
2. The variance of the response variable, denoted σ2, is the same for all of the populations.
3. The observations must be independent.
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password-protected website or school-approved learning management system for classroom use.
27
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
Sampling distribution of 𝑥,ҧ given 𝐻0 is true (𝐻0 : 𝜇1 = 𝜇2 = 𝜇3 = 𝜇).
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28
Statistics for Business and Economics (14e)
Analysis of Variance: A Conceptual Overview
Sampling distribution of 𝑥,ҧ given 𝐻0 is false.
𝜇
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29
Statistics for Business and Economics (14e)
Analysis of Variance under the Completely Randomized Design
Step 1. Calculate Between-Treatments Estimate of Population Variance (MSTR)
Step 2. Calculate Within-Treatments Estimate of Population Variance (MSE)
Step 3. Calculate Comparing the Variance Estimates: The F Test (= MSTR/MSE)
Step 4. Build ANOVA Table
Step 5. Test Hypothesis with p-value
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30
Statistics for Business and Economics (14e)
Step 1. Between-Treatments Estimate of Population Variance σ2
The estimate of σ2 based on the variation of the sample means is called the
mean square due to treatments and is denoted by MSTR.
=
𝑆𝑆𝑇𝑅
𝑘−1
where Numerator is called the sum of squares due to treatments (SSTR),
Denominator is the degrees of freedom associated with SSTR, and
𝑘
is the number of groups.
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31
Statistics for Business and Economics (14e)
Step 1. Between-Treatments Estimate of Population Variance σ2
=
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password-protected website or school-approved learning management system for classroom use.
𝑆𝑆𝑇𝑅
𝑘−1
32
Statistics for Business and Economics (14e)
Step 1. Between-Treatments Estimate of Population Variance σ2
=
𝑆𝑆𝑇𝑅
𝑘−1
𝑘=3
𝑆𝑆𝑇𝑅 = ෍ 𝑛𝑗 𝑥𝑗ҧ − 𝑥Ӗ
2
𝑗=1
= 5 62 − 60
= 520
2
+ 5 66 − 60
2
+ 5 52 − 60
𝑆𝑆𝑇𝑅
520
𝑀𝑆𝑇𝑅 =
=
= 260
𝑘−1 3−1
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33
2
Statistics for Business and Economics (14e)
Step 2. Within-Treatments Estimate of Population Variance σ2
The estimate of σ2 based on the variation of the sample observations within each
sample is called the mean square error and is denoted by MSE.
=
𝑆𝑆𝐸
𝑛𝑇 −𝑘
where Numerator is called the sum of squares due to error (SSE),
Denominator is the degrees of freedom associated with SSE, and
𝑛 𝑇 is total sample size of 𝑘 groups.
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34
Statistics for Business and Economics (14e)
Step 2. Within-Treatments Estimate of Population Variance σ2
=
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password-protected website or school-approved learning management system for classroom use.
𝑆𝑆𝐸
𝑛𝑇 −𝑘
35
Statistics for Business and Economics (14e)
Step 2. Within-Treatments Estimate of Population Variance σ2
=
𝑆𝑆𝐸
𝑛𝑇 −𝑘
𝑘=3
𝑆𝑆𝐸 = ෍ 𝑛𝑗 − 1 𝑠𝑗2
𝑗=1
= 5 − 1 27.5 + 5 − 1 26.5 + 5 − 1 31
= 340
𝑆𝑆𝐸
340
𝑀𝑆𝐸 =
=
= 28.33
𝑛 𝑇 − 𝑘 15 − 3
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36
Statistics for Business and Economics (14e)
Step 3. Comparing the Variance Estimates: The F Test
• If the null hypothesis is true and the ANOVA assumptions are valid, the sampling
distribution of MSTR/MSE is an 𝐹 distribution with MSTR degrees of freedom equal
to 𝑘 – 1 and MSE degrees of freedom equal to
• If the means of the 𝑘 populations are not equal, the value of MSTR/MSE will be
inflated because MSTR overestimates σ2.
• Hence, we will reject 𝐻0 if the resulting value of MSTR/MSE appears to be too large to
have been selected at random from the appropriate 𝐹 distribution.
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37
Statistics for Business and Economics (14e)
Step 3. Comparing the Variance Estimates: The F Test
• If the null hypothesis is true and the ANOVA assumptions are valid, the sampling
distribution of MSTR/MSE is an 𝐹 distribution with MSTR degrees of freedom equal
𝑴𝑺𝑻𝑹
to 𝑘 – 1 and MSE degrees of freedom equal to
~𝑭𝒌−𝟏,𝒏𝑻 −𝒌
𝑴𝑺𝑬
• If the means of the 𝑘 populations are not equal, the value of MSTR/MSE will be
inflated because MSTR overestimates σ2.
• Hence, we will reject 𝐻0 if the resulting value of MSTR/MSE appears to be too large to
have been selected at random from the appropriate 𝐹 distribution.
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38
Statistics for Business and Economics (14e)
Step 3. Comparing the Variance Estimates: The F Test
• If the null hypothesis is true and the ANOVA assumptions are valid, the sampling
distribution of MSTR/MSE is an 𝐹 distribution with MSTR degrees of freedom equal
𝑴𝑺𝑻𝑹
to 𝑘 – 1 and MSE degrees of freedom equal to
~𝑭𝒌−𝟏,𝒏𝑻 −𝒌
𝑴𝑺𝑬
• If the means of the 𝑘 populations are not equal, the value of MSTR/MSE will be
inflated because MSTR overestimates σ2.
(MSTR is large)
• Hence, we will reject 𝐻0 if the resulting value of MSTR/MSE appears to be too large to
have been selected at random from the appropriate 𝐹 distribution.
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39
Statistics for Business and Economics (14e)
Step 4. ANOVA Table for a Completely Randomized Design
SST is partitioned into SSTR and SSE.
SST’s degrees of freedom (df) are partitioned into SSTR’s df and SSE’s df.
Source of Variation
Sum of Squares
Degrees of Freedom
Mean Square
F
p-Value
Treatments
SSTR
K minus 1
Begin equation. MSTR
equals Start fraction,
SSTR over k minus 1.
End fraction. End
equation.
Begin fraction. MSTR
over MSE. End
fraction.
empty cell
Error
SSE
N subscript T baseline
minus k
Begin equation. MSE
equals start fraction
SSE over n subscript T
baseline minus k end
fraction. End
equation.
empty cell
empty cell
Total
SST
N subscript baseline
minus 1
empty cell
empty cell
empty cell
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40
Statistics for Business and Economics (14e)
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41
Statistics for Business and Economics (14e)
Step 5. Test for the Equality of 𝑘 Population Means
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42
Statistics for Business and Economics (14e)
Step 5. Test for the Equality of 𝑘 Population Means
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43
Statistics for Business and Economics (14e)
Step 5. Test for the Equality of 𝑘 Population Means
Critical value
approach
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P-value approach
44
Statistics for Business and Economics (14e)
P-value approach:
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password-protected website or school-approved learning management system for classroom use.
45
Statistics for Business and Economics (14e)
P-value approach:
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password-protected website or school-approved learning management system for classroom use.
46
Statistics for Business and Economics (14e)
Step 5. Comparing the Variance Estimates: The F Test
Critical value approach:
Sampling Distribution of MSTR/MSE
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47
Statistics for Business and Economics (14e)
Step 5. Comparing the Variance Estimates: The F Test
Critical value approach:
Sampling Distribution of MSTR/MSE
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password-protected website or school-approved learning management system for classroom use.
48
Statistics for Business and Economics (14e)
Step 5. Comparing the Variance Estimates: The F Test
Critical value approach:
Sampling Distribution of MSTR/MSE
if 𝐹 ≥ 𝐹𝛼
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49
Statistics for Business and Economics (14e)
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password-protected website or school-approved learning management system for classroom use.
50
Statistics for Business and Economics (14e)
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password-protected website or school-approved learning management system for classroom use.
51
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
AutoShine, Inc. is considering marketing a long- lasting car wax. Three different waxes (Type 1, Type
2, and Type 3) have been developed. In order to test the durability of these waxes, 5 new cars were
waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an
automatic carwash until the wax coating showed signs of deterioration.
The number of times each car went through the carwash before its wax deteriorated is shown on the
next slide. AutoShine, Inc. must decide which wax to market. Are the three waxes equally effective?
Factor . . . Car wax
Treatments . . . Type I, Type 2, Type 3
Experimental units . . . Cars
Response variable . . . Number of washes
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password-protected website or school-approved learning management system for classroom use.
52
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
Wax
Type 1
Wax
Type 2
Wax
Type 3
1
27
33
29
2
30
28
28
3
29
31
30
4
28
30
32
5
31
30
31
Sample Mean
29.0
30.4
30.0
Sample Variance
2.5
3.3
2.5
Observation
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53
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
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54
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
Mean Square Between Treatments: (Because the sample sizes are all equal)
Mean Square Error:
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55
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
Rejection Rule:
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56
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
Test Statistic:
Conclusion:
There is insufficient evidence to conclude that the mean number of washes
for the three wax types are not all the same.
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57
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: A Completely
Randomized Design
ANOVA Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
p-Value
Treatments
5.2
2
2.60
0.939
0.42
Error
33.2
12
2.77
Total
38.4
14
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58
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
Example: Reed Manufacturing
Janet Reed would like to know if there is any significant difference in the mean number of
hours worked per week for the department managers at her three manufacturing plants (in
Buffalo, Pittsburgh, and Detroit). An 𝐹 test will be conducted using α = 0.05.
A simple random sample of five managers from each of the three plants was taken and the
number of hours worked by each manager in the previous week is shown on the next slide.
Factor . . . Manufacturing plant
Treatments . . . Buffalo, Pittsburgh, Detroit
Experimental units . . . Managers
Response variable . . . Number of hours worked
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59
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
Observation
Plant 1 Buffalo
Plant 2 Pittsburgh
Plant 3 Detroit
1
48
73
51
2
54
63
63
3
57
66
61
4
54
64
54
5
62
74
56
55
68
57
26.0
26.5
24.5
Sample Mean
Sample Variance
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60
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
1. Develop the hypotheses.
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61
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
2. Specify the level of significance. α = 0.05
3. Compute the value of the test statistic.
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62
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
3. Compute the value of the test statistic.
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password-protected website or school-approved learning management system for classroom use.
63
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
ANOVA Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean Square
F
p-Value
Treatment
490
2
245
9.55
.0033
Error
308
12
25.667
EMPTY CELL
EMPTY CELL
Total
798
14
EMPTY CELL
EMPTY CELL
EMPTY CELL
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64
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
p-value approach
4. Compute the p –value.
With 2 numerator df and 12 denominator df, the p-value is 0.01 for 𝐹 = 6.93.
Therefore, the p-value is less than 0.01 for 𝐹 = 9.55.
5.
We can conclude that the mean number of hours worked per week by
department managers is not the same at all 3 plants.
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65
Statistics for Business and Economics (14e)
Testing for the Equality of 𝑘 Population Means: An Observational Study
Critical Value Approach
4. Determine the critical value and rejection rule.
5.
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66
Statistics for Business and Economics (14e)
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password-protected website or school-approved learning management system for classroom use.
67
Statistics for Business and Economics (14e)
Multiple Comparison Procedures
• Suppose that analysis of variance has provided statistical evidence to reject the
null hypothesis of equal population means.
• Fisher’s least significant difference (LSD) procedure can be used to determine
where the differences occur.
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68
Statistics for Business and Economics (14e)
(𝑥)ҧ
ҧ
𝑥=60
• Do the means of population A and B differ ?
• Do the means of population A and C differ ?
• Do the means of population B and C differ ?
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69
Statistics for Business and Economics (14e)
(𝑥)ҧ
ҧ
𝑥=60
• Do the means of population A and B differ ? 𝐻0 : 𝜇𝐴 = 𝜇𝐵 𝑣𝑠 𝐻𝑎 : 𝜇𝐴 ≠ 𝜇𝐵
• Do the means of population A and C differ ? 𝐻0 : 𝜇𝐴 = 𝜇𝐶 𝑣𝑠 𝐻𝑎 : 𝜇𝐴 ≠ 𝜇𝐶
• Do the means of population B and C differ ? 𝐻0 : 𝜇𝐵 = 𝜇𝐶 𝑣𝑠 𝐻𝑎 : 𝜇𝐵 ≠ 𝜇𝐶
© 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a
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70
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure
1. Hypotheses:
where i=A, B; j=B, C
2. Test Statistic:
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71
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure
Chapter 10. Hypothesis Tests About μ1 – μ2
when σ1 and σ2 are Unknown
1. Hypotheses:
where i=A, B; j=B, C
2. Test Statistic:
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72
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure
Chapter 10. Hypothesis Tests About μ1 – μ2
when σ1 and σ2 are Unknown
1. Hypotheses:
where i=A, B; j=B, C
2. Test Statistic:
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73
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure
3. Rejection Rule:
𝑛 𝑇 is the total number of sample;
k is number of groups/methods/treatments.
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74
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure Based on the Test Statistic 𝑥ҧ𝑖 − 𝑥𝑗ҧ
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75
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure Based on the Test Statistic 𝑥ҧ𝑖 − 𝑥𝑗ҧ
1
5
1
5
= 2.179 28.33( + )≈ 7.304
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76
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure Based on the Test Statistic 𝑥ҧ𝑖 − 𝑥𝑗ҧ
1. LSD for Treatments A and B
• Hypothesis 1: 𝐻0 : 𝜇𝐴 = 𝜇𝐵 𝑣𝑠 𝐻𝑎 : 𝜇𝐴 ≠ 𝜇𝐵
• Rejection Rule: Reject 𝐻0 if 𝑥𝐴ҧ − 𝑥ҧ𝐵 ≥ 7.304
• Test Statistic: 𝑥𝐴ҧ − 𝑥ҧ𝐵 = 62 − 66 = 4
• Conclusion: The mean number of units produced at Treatment A is equal to the mean
number produced at Treatment B.
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77
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure Based on the Test Statistic 𝑥ҧ𝑖 − 𝑥𝑗ҧ
2. LSD for Treatments A and C
• Hypothesis 1: 𝐻0 : 𝜇𝐴 = 𝜇𝐶 𝑣𝑠 𝐻𝑎 : 𝜇𝐴 ≠ 𝜇𝐶
• Rejection Rule: Reject 𝐻0 if 𝑥𝐴ҧ − 𝑥ҧ𝐶 ≥ 7.304
• Test Statistic: 𝑥𝐴ҧ − 𝑥ҧ𝐶 = 62 − 52 = 10
• Conclusion: The mean number of units produced at Treatment A is not equal to the mean
number produced at Treatment C.
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78
Statistics for Business and Economics (14e)
Fisher’s LSD Procedure Based on the Test Statistic 𝑥ҧ𝑖 − 𝑥𝑗ҧ
3. LSD for Treatments B and C
• Hypothesis 1: 𝐻0 : 𝜇𝐵 = 𝜇𝐶 𝑣𝑠 𝐻𝑎 : 𝜇𝐵 ≠ 𝜇𝐶
• Rejection Rule: Reject 𝐻0 if 𝑥ҧ𝐵 − 𝑥ҧ𝐶 ≥ 7.304
• Test Statistic: 𝑥ҧ𝐵 − 𝑥ҧ𝐶 = 66 − 52 = 14
• Conclusion: The mean number of units produced at Treatment B is not equal to the
mean number produced at Treatment C.
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79
Statistics for Business and Economics (14e)
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80