Title: Statistics for Business and Economics
1Statistics for Business and Economics
- Chapter 6
- Inferences Based on a Single Sample Tests of
Hypothesis
2Learning 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
3Statistical Methods
StatisticalMethods
InferentialStatistics
DescriptiveStatistics
HypothesisTesting
Estimation
4Hypothesis Testing Concepts
5Hypothesis Testing
6Whats 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!
7Null 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
8Alternative Hypothesis
- Opposite of null hypothesis
- Always has inequality sign ?,??, or ?
- Designated Ha
- Specified Ha ? ?,??, or ? some value
- Example, Ha ? lt 3
9Identifying 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)
10What 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
11What 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
12What 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
13What 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
14Basic Idea
... therefore, we reject the hypothesis that ?
50.
15Level 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
16Rejection Region (One-Tail Test)
17Rejection Region (One-Tail Test)
Sampling Distribution
Level of Confidence
18Rejection Regions (Two-Tailed Test)
19Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
20Rejection Regions (Two-Tailed Test)
Sampling Distribution
Level of Confidence
21Decision Making Risks
22Errors 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)
23Decision Results
H0 Innocent
Jury Trial
Actual Situation
Verdict
Innocent
Guilty
Innocent
Correct
Error
Guilty
Error
Correct
24? ? Have an Inverse Relationship
?
?
25Factors 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
26Hypothesis Testing Steps
27H0 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
28One Population Tests
29Two-Tailed Z Test of Mean (? Known)
30One Population Tests
31Two-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
32Two-Tailed Z Test for Mean Hypotheses
H0????0 Ha ??? 0
Reject H
Reject H
0
a / 2
a / 2
Z
0
33Two-Tailed Z Test Finding Critical Z
What is Z given ? .05?
s
1
? / 2 .025
?
Z
0
34Two-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.
35Two-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
36Two-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?
37Two-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
38One-Tailed Z Test of Mean (? Known)
39One-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
40One-Tailed Z Test for Mean Hypotheses
H0????0 Ha ??lt 0
41One-Tailed Z Test Finding Critical Z
What Is Z given ? .025?
s
1
? .025
?
Z
0
42One-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.
43One-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
44One-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?
45One-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
46Observed Significance Levels p-Values
47p-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
48Two-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.
49Two-Tailed Z Test p-Value Solution
Z value of sample statistic (observed)
?
50Two-Tailed Z Test p-Value Solution
p-value is P(Z ? -1.50 or Z ? 1.50)
Z value of sample statistic (observed)
?
51Two-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
?
?
52Two-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
53One-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.
54One-Tailed Z Test p-Value Solution
Z value of sample statistic
?
55One-Tailed Z Test p-Value Solution
p-Value is P(Z ? 1.50)
?
Use alternative hypothesis to find direction
Z value of sample statistic
?
56One-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
?
57One-Tailed Z Test p-Value Solution
p-Value .0668
Reject H0
? .05
Z
0
1.50
58p-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)?
59p-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
?
60Two-Tailed t Test of Mean (? Unknown)
61One Population Tests
62t 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
63Two-Tailed t Test Finding Critical t Values
Given n 3 ? .10
64Two-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.
65Two-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
66Two-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?
67Two-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
68One-Tailed t Test of Mean (? Unknown)
69One-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.
70One-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
71One-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)?
72One-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
73Z Test of Proportion
74Data Types
75Qualitative 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?
76Proportions
- Involve qualitative variables
- Fraction or percentage of population in a
category - If two qualitative outcomes, binomial
distribution - Possess or dont possess characteristic
77Sampling 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
78Standardizing Sampling Distribution of Proportion
Sampling Distribution
Standardized Normal Distribution
?
79One Population Tests
80One-Sample Z Test for Proportion
2. Z-test statistic for proportion
81One-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.
82One-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
83One-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?
84One-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
85Calculating Type II Error Probabilities
86Power 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
87Finding PowerStep 1
88Finding PowerSteps 2 3
89Finding PowerStep 4
90Finding PowerStep 5
?
True Situation ?a 360 (Ha)
Draw
? .154
?
?
?
1-? .846
Specify
Z Table
363.065
?
X
?
360
a
91Power 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
92Chi-Square (?2) Test of Variance
93One Population Tests
94Chi-Square (?2) Testfor Variance
- Tests one population variance or standard
deviation - Assumes population is approximately normally
distributed - Null hypothesis is H0 ?2 ?02
95Chi-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
96Finding Critical Value Example
What is the critical ?2 value givenHa ?2 gt
0.7n 3? .05?
df n - 1 2
97Finding Critical Value Example
What is the critical ?2 value givenHa ?2 lt
0.7n 3? .05?
98Finding Critical Value Example
What is the critical ?2 value givenHa ?2 lt
0.7n 3? .05?
99Chi-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.
100Chi-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
101Conclusion
- 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