Statistics for Business and Economics

1
Statistics for Business and Economics

• Chapter 6
• Inferences Based on a Single Sample Tests of
  Hypothesis

2
Learning Objectives

• Distinguish Types of Hypotheses
• Describe Hypothesis Testing Process
• Explain p-Value Concept
• Solve Hypothesis Testing Problems Based on a
  Single Sample
• Explain Power of a Test

3
Statistical Methods
StatisticalMethods
InferentialStatistics
DescriptiveStatistics
HypothesisTesting
Estimation
4
Hypothesis Testing Concepts
5
Hypothesis Testing
6
Whats a Hypothesis?

• A belief about a population parameter
• Parameter is population mean, proportion,
  variance
• Must be stated before analysis

I believe the mean GPA of this class is 3.5!
7
Null Hypothesis

• What is tested
• Has serious outcome if incorrect decision made
• Always has equality sign ?, ?, or ??
• Designated H0 (pronounced H-oh)
• Specified as H0 ? ? some numeric value
• Specified with sign even if ? or ??
• Example, H0 ? ? 3

8
Alternative Hypothesis

• Opposite of null hypothesis
• Always has inequality sign ?,??, or ?
• Designated Ha
• Specified Ha ? ?,??, or ? some value
• Example, Ha ? lt 3

9
Identifying HypothesesSteps

• Example problem Test that the population mean is
  not 3
• Steps
• State the question statistically (? ? 3)
• State the opposite statistically (? 3)
• Must be mutually exclusive exhaustive
• Select the alternative hypothesis (? ? 3)
• Has the ?, lt, or gt sign
• State the null hypothesis (? 3)

10
What Are the Hypotheses?

Is the population average amount of TV viewing 12
hours?

• State the question statistically ? 12
• State the opposite statistically ? ? 12
• Select the alternative hypothesis Ha ? ? 12
• State the null hypothesis H0 ? 12

11
What Are the Hypotheses?

Is the population average amount of TV viewing
different from 12 hours?

• State the question statistically ? ? 12
• State the opposite statistically ? 12
• Select the alternative hypothesis Ha ? ? 12
• State the null hypothesis H0 ? 12

12
What Are the Hypotheses?

Is the average cost per hat less than or equal to
20?

• State the question statistically ? ? 20
• State the opposite statistically ? ? 20
• Select the alternative hypothesis Ha ? ? 20
• State the null hypothesis H0 ? ? 20

13
What Are the Hypotheses?

Is the average amount spent in the bookstore
greater than 25?

• State the question statistically ? ? 25
• State the opposite statistically ? ? 25
• Select the alternative hypothesis Ha ? ? 25
• State the null hypothesis H0 ? ? 25

14
Basic Idea
... therefore, we reject the hypothesis that ?
50.
15
Level of Significance

• Probability
• Defines unlikely values of sample statistic if
  null hypothesis is true
• Called rejection region of sampling
  distribution
• Designated ??(alpha)
• Typical values are .01, .05, .10
• Selected by researcher at start

16
Rejection Region (One-Tail Test)
17
Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
18
Rejection Regions (Two-Tailed Test)
19
Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
20
Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
21
Decision Making Risks
22
Errors in Making Decision

• Type I Error
• Reject true null hypothesis
• Has serious consequences
• Probability of Type I Error is ??(alpha)
• Called level of significance
• Type II Error
• Do not reject false null hypothesis
• Probability of Type II Error is ??(beta)

23
Decision Results
H0 Innocent
Jury Trial
Actual Situation
Verdict
Innocent
Guilty
Innocent
Correct
Error
Guilty
Error
Correct
24
? ? Have an Inverse Relationship
?
?
25
Factors Affecting ?

• True value of population parameter
• Increases when difference with hypothesizedparame
  ter decreases
• Significance level, ?
• Increases when ??decreases
• Population standard deviation, ?
• Increases when ? increases
• Sample size, n
• Increases when n decreases

26
Hypothesis Testing Steps
27
H0 Testing Steps

• State H0
• State Ha
• Choose ?
• Choose n
• Choose test

• Set up critical values
• Collect data
• Compute test statistic
• Make statistical decision
• Express decision

28
One Population Tests
29
Two-Tailed Z Test of Mean (? Known)
30
One Population Tests
31
Two-Tailed Z Test for Mean (? Known)

• Assumptions
• Population is normally distributed
• If not normal, can be approximated by normal
  distribution (n ? 30)
• Alternative hypothesis has ? sign

3. Z-Test Statistic
32
Two-Tailed Z Test for Mean Hypotheses
H0????0 Ha ??? 0
Reject H
Reject H
0
a / 2
a / 2
Z
0
33
Two-Tailed Z Test Finding Critical Z
What is Z given ? .05?
s
1
? / 2 .025
?
Z
0
34
Two-Tailed Z Test Example

• Does an average box of cereal contain 368 grams
  of cereal? A random sample of 25 boxes showed x
  372.5. The company has specified ? to be 25
  grams. Test at the .05 level of significance.

368 gm.
35
Two-Tailed Z Test Solution

• H0
• Ha
• ? ?
• n ?
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
No evidence average is not 368
36
Two-Tailed Z Test Thinking Challenge

• Youre a Q/C inspector. You want to find out if
  a new machine is making electrical cords to
  customer specification average breaking strength
  of 70 lb. with ? 3.5 lb. You take a sample of
  36 cords compute a sample mean of 69.7 lb. At
  the .05 level of significance, is there evidence
  that the machine is not meeting the average
  breaking strength?

37
Two-Tailed Z Test Solution

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
No evidence average is not 70
38
One-Tailed Z Test of Mean (? Known)
39
One-Tailed Z Test for Mean (? Known)

• Assumptions
• Population is normally distributed
• If not normal, can be approximated by normal
  distribution (n ? 30)
• Alternative hypothesis has lt or gt sign

3. Z-test Statistic
40
One-Tailed Z Test for Mean Hypotheses
H0????0 Ha ??lt 0
41
One-Tailed Z Test Finding Critical Z
What Is Z given ? .025?
s
1
? .025
?
Z
0
42
One-Tailed Z Test Example

• Does an average box of cereal contain more than
  368 grams of cereal? A random sample of 25 boxes
  showed x 372.5. The company has specified ?
  to be 25 grams. Test at the .05 level of
  significance.

368 gm.
43
One-Tailed Z Test Solution

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
No evidence average is more than 368
44
One-Tailed Z Test Thinking Challenge

• Youre an analyst for Ford. You want to find out
  if the average miles per gallon of Escorts is at
  least 32 mpg. Similar models have a standard
  deviation of 3.8 mpg. You take a sample of 60
  Escorts compute a sample mean of 30.7 mpg. At
  the .01 level of significance, is there evidence
  that the miles per gallon is at least 32?

45
One-Tailed Z Test Solution

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
Reject at ? .01
There is evidence average is less than 32
46
Observed Significance Levels p-Values
47
p-Value

• Probability of obtaining a test statistic more
  extreme (??or ???than actual sample value, given
  H0 is true
• Called observed level of significance
• Smallest value of ? for which H0 can be rejected
• Used to make rejection decision
• If p-value ? ?, do not reject H0
• If p-value lt ?, reject H0

48
Two-Tailed Z Test p-Value Example

• Does an average box of cereal contain 368 grams
  of cereal? A random sample of 25 boxes showed x
  372.5. The company has specified ? to be 25
  grams. Find the p-Value.

368 gm.
49
Two-Tailed Z Test p-Value Solution
Z value of sample statistic (observed)
?
50
Two-Tailed Z Test p-Value Solution
p-value is P(Z ? -1.50 or Z ? 1.50)
Z value of sample statistic (observed)
?
51
Two-Tailed Z Test p-Value Solution
p-value is P(Z ? -1.50 or Z ? 1.50) .1336
?
1/2 p-Value
1/2 p-Value
.5000- .4332 .0668
.0668
.0668
Z
0
1.50
-1.50
Z value of sample statistic
From Z table lookup 1.50
?
?
52
Two-Tailed Z Test p-Value Solution
1/2 p-Value .0668
1/2 p-Value .0668
Reject H0
Reject H0
1/2 ? .025
1/2 ? .025
Z
0
1.50
-1.50
53
One-Tailed Z Test p-Value Example

• Does an average box of cereal contain more than
  368 grams of cereal? A random sample of 25 boxes
  showed x 372.5. The company has specified ? to
  be 25 grams. Find the p-Value.

368 gm.
54
One-Tailed Z Test p-Value Solution
Z value of sample statistic
?
55
One-Tailed Z Test p-Value Solution
p-Value is P(Z ? 1.50)
?
Use alternative hypothesis to find direction
Z value of sample statistic
?
56
One-Tailed Z Test p-Value Solution
p-Value is P(Z ? 1.50) .0668
?
p-Value
?
Use alternative hypothesis to find direction
.0668
.5000- .4332 .0668
.4332
?
Z value of sample statistic
From Z table lookup 1.50
?
57
One-Tailed Z Test p-Value Solution
p-Value .0668
Reject H0
? .05
Z
0
1.50
58
p-Value Thinking Challenge

• Youre an analyst for Ford. You want to find out
  if the average miles per gallon of Escorts is at
  least 32 mpg. Similar models have a standard
  deviation of 3.8 mpg. You take a sample of 60
  Escorts compute a sample mean of 30.7 mpg.
  What is the value of the observed level of
  significance (p-Value)?

59
p-Value Solution
p-Value is P(Z ? -2.65) .004.p-Value lt (?
.01). Reject H0.
Z
0
-2.65
Z value of sample statistic
?
60
Two-Tailed t Test of Mean (? Unknown)
61
One Population Tests
62
t Test for Mean (? Unknown)

• Assumptions
• Population is normally distributed
• If not normal, only slightly skewed large
  sample (n ? 30) taken
• Parametric test procedure
• t test statistic

63
Two-Tailed t Test Finding Critical t Values
Given n 3 ? .10
64
Two-Tailed t Test Example

• Does an average box of cereal contain 368 grams
  of cereal? A random sample of 36 boxes had a
  mean of 372.5 and a standard deviation of 12
  grams. Test at the .05 level of significance.

368 gm.
65
Two-Tailed t Test Solution

• H0
• Ha
• ?
• df
• Critical Value(s)

Test Statistic Decision Conclusion
Reject at ? .05
There is evidence population average is not 368
66
Two-Tailed t TestThinking Challenge

• You work for the FTC. A manufacturer of
  detergent claims that the mean weight of
  detergent is 3.25 lb. You take a random sample
  of 64 containers. You calculate the sample
  average to be 3.238 lb. with a standard deviation
  of .117 lb. At the .01 level of significance, is
  the manufacturer correct?

67
Two-Tailed t Test Solution

• H0
• Ha
• ? ?
• df ?
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .01
There is no evidence average is not 3.25
68
One-Tailed t Test of Mean (? Unknown)
69
One-Tailed t TestExample

• Is the average capacity of batteries at least 140
  ampere-hours? A random sample of 20 batteries
  had a mean of 138.47 and a standard deviation of
  2.66. Assume a normal distribution. Test at the
  .05 level of significance.

70
One-Tailed t Test Solution

• H0
• Ha
• ?
• df
• Critical Value(s)

Test Statistic Decision Conclusion
Reject at ? .05
There is evidence population average is less than
140
71
One-Tailed t Test Thinking Challenge

• Youre a marketing analyst for Wal-Mart.
  Wal-Mart had teddy bears on sale last week. The
  weekly sales ( 00) of bears sold in 10 stores
  was 8 11 0 4 7 8 10 5 8 3 At
  the .05 level of significance, is there evidence
  that the average bear sales per store is more
  than 5 ( 00)?

72
One-Tailed t Test Solution

• H0
• Ha
• ?
• df
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
There is no evidence average is more than 5
73
Z Test of Proportion
74
Data Types
75
Qualitative Data

• Qualitative random variables yield responses
  that classify
• e.g., Gender (male, female)
• Measurement reflects number in category
• Nominal or ordinal scale
• Examples
• Do you own savings bonds?
• Do you live on-campus or off-campus?

76
Proportions

• Involve qualitative variables
• Fraction or percentage of population in a
  category
• If two qualitative outcomes, binomial
  distribution
• Possess or dont possess characteristic

77
Sampling Distribution of Proportion

• Approximated by Normal Distribution
• Excludes 0 or n
• Mean
• Standard Error

Sampling Distribution

P(P
)
.3
.2
.1

P
.0
.0
.2
.4
.6
.8
1.0
where p0 Population Proportion
78
Standardizing Sampling Distribution of Proportion
Sampling Distribution
Standardized Normal Distribution
?
79
One Population Tests
80
One-Sample Z Test for Proportion
2. Z-test statistic for proportion
81
One-Proportion Z Test Example

• The present packaging system produces 10
  defective cereal boxes. Using a new system, a
  random sample of 200 boxes had?11 defects. Does
  the new system produce fewer defects? Test at
  the .05 level of significance.

82
One-Proportion Z Test Solution

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
Reject at ? .05
There is evidence new system lt 10 defective
83
One-Proportion Z Test Thinking Challenge

• Youre an accounting manager. A year-end audit
  showed 4 of transactions had errors. You
  implement new procedures. A random sample of 500
  transactions had 25 errors. Has the proportion
  of incorrect transactions changed at the .05
  level of significance?

84
One-Proportion Z Test Solution

• H0
• Ha
• ?
• n
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
There is evidence proportion is not 4
85
Calculating Type II Error Probabilities
86
Power of Test

• Probability of rejecting false H0
• Correct decision
• Designated 1 - ?
• Used in determining test adequacy
• Affected by
• True value of population parameter
• Significance level ?
• Standard deviation sample size n

87
Finding PowerStep 1
88
Finding PowerSteps 2 3
89
Finding PowerStep 4
90
Finding PowerStep 5
?
True Situation ?a 360 (Ha)
Draw
? .154
?
?
?
1-? .846
Specify
Z Table
363.065
?
X
?
360
a
91
Power Curves
H0 ? ???0
H0 ? ???0
Power
Power
Possible True Values for ?a
Possible True Values for ?a
H0 ? ??0
Power
?? 368 in Example
Possible True Values for ?a
92
Chi-Square (?2) Test of Variance
93
One Population Tests
94
Chi-Square (?2) Testfor Variance

• Tests one population variance or standard
  deviation
• Assumes population is approximately normally
  distributed
• Null hypothesis is H0 ?2 ?02

95
Chi-Square (?2) Distribution
Select simple random
Population
sample, size n.
Sampling Distributions
Compute

s
2
for Different Sample
Sizes
s
m
c

s
2
2
2
Compute

(n-1)s
/
c
2
1
2
3
0
Astronomical number
c
of
2

values
96
Finding Critical Value Example
What is the critical ?2 value givenHa ?2 gt
0.7n 3? .05?
df n - 1 2
97
Finding Critical Value Example
What is the critical ?2 value givenHa ?2 lt
0.7n 3? .05?
98
Finding Critical Value Example
What is the critical ?2 value givenHa ?2 lt
0.7n 3? .05?
99
Chi-Square (?2) Test Example

• Is the variation in boxes of cereal, measured by
  the variance, equal to 15 grams? A random sample
  of 25 boxes had a standard deviation of 17.7
  grams. Test at the .05 level of significance.

100
Chi-Square (?2) Test Solution

• H0
• Ha
• ?
• df
• Critical Value(s)

Test Statistic Decision Conclusion
Do not reject at ? .05
There is no evidence ?2 is not 15
101
Conclusion

• Distinguished Types of Hypotheses
• Described Hypothesis Testing Process
• Explained p-Value Concept
• Solved Hypothesis Testing Problems Based on a
  Single Sample
• Explained Power of a Test