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Hypothesis Testing
Chapter 10
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 1
Hypothesis Testing Basics
 A hypothesis is a statistical assertion
concerning one or more populations.
 Null hypothesis: A hypothesis to be tested. We
use the symbol H0 to represent the null
hypothesis
 Alternative hypothesis: A hypothesis to be
considered as an alternative to the null
hypothesis. We use the symbol H1 to represent
the alternative hypothesis.

- The alternative hypothesis is the one
believed to be true, or what you are trying to
prove is true.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 2
Practical Significance vs. Statistical
Significance
 When no practical difference exist, it may be
possible to detect a statistically significant
difference
 Hypothesis tests are performed to determine if a
claim has significant statistical merit
 Although a hypothesis claims may be found
statistically significant, the effort or expense to
implement any changes may not be worth it.
 For example if a study showed that a budget
helps people save an extra $10 per year, a
budget that only saves $10 extra per year does
not have any practical significance.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 3
Statistical Hypothesis Testing
 In statistics, a hypothesis test is conducted on a set
of two mutually exclusive statements:
H0 : null hypothesis
H1 : alternate hypothesis
 Example
H0 : μ = 17
H1 : μ ≠ 17
 We sometimes refer to the null hypothesis as the
“equals” hypothesis.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 4
Hypothesis Testing Basics
 Null hypothesis must be accepted (fail to reject) or
rejected
 Test Statistic: A value which functions as the decision
maker. The decision to “reject” or “fail to reject” is based
on information contained in a sample drawn from the
population of interest.
 Rejection region: If test statistic falls in some interval
which support alternative hypothesis, we reject the null
hypothesis.
 Acceptance Region: It test statistic falls in some interval
which support null hypothesis, we fail to reject the null
hypothesis.
 Critical Value: The point which divide the rejection
region and acceptance
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 5
Hypothesis Testing Basics
Test statistic; n is large, standard deviation is
known
Z-statistic
Test statistic: n is small, standard deviation
is unknown
T-statistic
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 6
Tests of Hypotheses - Graphics I
 We can make a decision about our hypotheses
based on our understanding of probability.
 We can visualize this probability by defining a
rejection region on the probability curve.
 The general location of the rejection region is
determined by the alternate hypothesis.
H0 : μ = _____
H1 : μ < _____
One-sided
MDH Ch10 Lecture 1 9th ed. v Spring
2015
H0 : μ = _____
H1 : μ ≠ _____
H0 : p = _____
H1 : p > _____
One-sided
Two-sided
EGR 252 2015
Slide 7
Choosing the Hypotheses
Your turn …
Suppose a coffee vending machine claims it
dispenses an 8-oz cup of coffee. You have been
using the machine for 6 months, but recently it
seems the cup isn’t as full as it used to be. You
plan to conduct a statistical hypothesis test. What
are your hypotheses?
H0 : μ = _____
H1 : μ ≠ _____
H0 : μ = _____
H1 : μ < _____
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 8
Potential errors in decision-making
H0 True
H0 False
Do not
reject H0
Correct
Decision
Type II
error
Reject H0
Type I
error
Correct
Decision
 α
 Probability of committing a
Type I error (incorrect
rejection of a true null
hypothesis)
 Probability of rejecting the
null hypothesis given that
the null hypothesis is true
 P (reject H0 | H0 is true)
MDH Ch10 Lecture 1 9th ed. v Spring
2015
 β
 Probability of committing a
Type II error. (failure to reject
a false null hypothesis)
 Power of the test = 1 - β
(probability of rejecting the
null hypothesis given that the
alternate is true.)
 Power = P (reject H0 | H1 is
true)
EGR 252 2015
Slide 9
Hypothesis Testing – Approach 1
 Approach 1 - Fixed probability of Type 1 error.
1. State the null and alternative hypotheses.
2. Choose a fixed significance level α.
3. Specify the appropriate test statistic and establish
the critical region based on α. Draw a graphic
representation.
4. Calculate the value of the test statistic based on
the sample data.
5. Make a decision to reject or fail to reject H0, based
on the location of the test statistic.
6. Make an engineering or scientific conclusion.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 10
Hypothesis Testing – Approach 2
p-value is a measure of the significance of your results
1. State the null and alternative hypotheses.
2. Choose an appropriate test statistic.
3. Calculate value of test statistic and determine pvalue. Draw a graphic representation.
4. Make a decision to reject or fail to reject H0, based
on the p-value by comparing it to a, the level of
significance . ***If not given assume a = 0.05
5. Make an engineering or scientific conclusion.
p-value < a
Reject Null
“p-value is low,
null must go”
p-value > a
Fail to Reject
Null (Accept)
“p-value is high,
null must fly”
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 11
Hypothesis Testing Tells Us …
Strong conclusion:
 If our calculated t-value is “outside” tα,ν (approach
1) or we have a small p-value (approach 2), then
we reject H0: μ = μ0 in favor of the alternate
hypothesis.
Weak conclusion:
 If our calculated t-value is “inside” tα,ν (approach 1)
or we have a “large” p-value (approach 2), then
we cannot reject H0: μ = μ0.
Failure to reject H0 does not imply that μ is
equal to the stated value (μ0), only that we do
not have sufficient evidence to support H1.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 12
Types of Tests
• Non-directional, two-tail test:
 H0: parameter = value
 H1: parameter ≠ value
• Directional, right-tail test:
 H0: parameter  value
 H1: parameter > value
• Directional, left-tail test:
 H0: parameter  value
 H1: parameter < value
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 13
Non-directional - Two-tail Test
Reject
Region
Reject
Region
Do Not
Reject H
Reject H
a/2
0
0
Reject H
1-a
–z
MDH Ch10 Lecture 1 9th ed. v Spring
2015
a/2
0
+z
EGR 252 2015
Slide 14
Directional-Right-tail Test
Reject
Region
Do Not Reject H
0
Reject H
1-a
0
a
+z
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 15
Directional-Left-tail Test
Reject
Region
Reject H
Do Not Reject H
0
1-a
a
0
–z
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 16
Example: Single Sample Test of the Mean
A sample of 20 cars driven under varying highway conditions
achieved fuel efficiencies as follows:
Sample mean
x = 34.271 mpg
Sample std dev
s = 2.915 mpg
Test the hypothesis that the population mean equals 35.0 mpg
vs. μ < 35.
Step 1: State the hypotheses.
H0: μ = 35
H1: μ < 35
Step 2: Determine the appropriate test statistic.
σ unknown, n = 20 Therefore, use t distribution
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 17
Example (concl.)
Approach 1: significance level (alpha)
Step 1: State hypotheses.
Step 2: Let’s set alpha at 0.05.
Step 3: Determine the critical value of t that separates the
reject H0 region from the do not reject H0 region.
ta, n-1 = t0.05,19 = 1.729
T
X -
S/ n
Since H1 format is “μ< μ0,” tcrit = -1.729
Step 4: tcalc = -1.11842
Step 5: Decision
Fail to reject H0
Step 6: Conclusion: The population mean is not significantly less
than 35 mpg.
****Do not conclude that the population mean equals 35 mpg.****
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 18
Single Sample Example (cont.)
Approach 2 p-value approach:
X -
T
S/ n
= -1.11842
Find probability from chart or use Excel’s tdist function.
P(x ≤ -1.118) = TDIST (1.118, 19, 1) = 0.139665
a = 0.05
a < p-value
P-value =
0.14
0______________________________1
Decision: Fail to reject null hypothesis (Accept)
Conclusion: The mean is not significantly less than 35 mpg.
MDH Ch10 Lecture 1 9th ed. v Spring
2015
EGR 252 2015
Slide 19