Download Outline Statistical Methods Hypothesis Testing Hypothesis Testing

Probability and Statistics
Outline
1. Distinguish Types of Hypotheses
LECTURE 8
INTRODUCTION TO HYPOTHESIS
TESTING
2. Describe the basics of hypothesis
testing
3. Explain the z-test for mean
Adapted from http://www.prenhall.com/mcclave
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Statistical Methods
Hypothesis Testing
Population
I believe the
population
mean age is 50
(hypothesis).
Reject
hypothesis!
Not close.



 


Random
sample
Mean
X = 20
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Hypothesis Testing
A hypothesis test allows us to draw
conclusions or make decisions
regarding population data from
sample data.
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What’s a Hypothesis?
Usually a statement about population
parameters

Parameter Is Population Mean,
Proportion, Variance

Must Be Stated
Before Analysis
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Example of hypothesis
The label on soft drink bottle states that it
contains 67.6 fluid ounces. Is there
evidence the label is incorrect?
What is the hypothesis in the above
context?
Research Hypothesis
•
•
•
•
•
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What we aim to show statistically
Statement we hope or suspect is true
Denoted as Ha or H1
Convention: no equality sign
Also called alternative hypothesis
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Null Hypothesis
•
Theory put forward because


•
•
•
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Believed to be true or used as starting point for
testing. So we assume Ho is true and use the
information specified in Ho as a starting point for
testing.
Not been proved
Opposite of Research Hypothesis
Convention: contain equality sign
Usually phrased as “no effect”, “no difference”
Called H0
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1.Example Problem: Test That the
Population Mean Is Not 3
2.Steps

State the Question Statistically (  3)
State the Opposite Statistically ( = 3)


Must Be Mutually Exclusive & Exhaustive
Select the Alternative Hypothesis (  3)


•
•
When doing exams, you will write Ho
before Ha. But to correctly figure out
hypotheses, you should write Ha first,
then the Ho. Normally, Ha contains the
question we wish to answer.
Take into account the convention
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Identifying Hypotheses
Steps

Setting up hypotheses
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
Has the , <, or > Sign
State the Null Hypothesis ( = 3)
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What Are the Hypotheses?
Is the population average amount of TV
viewing different from 12 hours?
What Are the Hypotheses?
Is the average cost per hat less than or
equal to $20?
State the question statistically:   12
State the question statistically:   20
State the opposite statistically:  = 12
Select the alternative hypothesis: Ha:  < 20
Select the alternative hypothesis: Ha:   12
State the null hypothesis: H0:   20
State the null hypothesis: H0:  = 12
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What Are the Hypotheses?
Is the average amount spent in the
bookstore greater than $25?
State the question statistically:   25
Jury trial example
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•
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Begin with idea the defendant is innocent
Collect data (evidence)
Convict if data are sufficiently inconsistent
with the initial idea
What should be our null hypothesis and
alternative hypothesis?
State the opposite statistically:   25
Select the alternative hypothesis: Ha:   25
•
State the null hypothesis: H0:   25
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Possible conclusions of a
hypothesis test
2 possible conclusions:
• Do not reject Ho: test is not significant
• Reject Ho: test is significant
Question: in the jury trial example, what
are the possible conclusions?
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Possible errors in testing
•
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Type I error: reject Ho if Ho is true
Type II error: do not reject Ho if Ho is
false
Question: identify Type I and Type II
errors in the jury trial example.
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Jury Trial Results
Possible errors in testing
H0: Innocent
•
Probability of type I error: alpha

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Significance level
Probability of type II error: beta
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Significance level
•
•
•
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Denoted a
Probability of making Type I error
Equals Total Area of Rejection Region
Should be decided by researcher at
start
•
Example Values Are .01, .05, .10, etc.
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If we aim to collect evidence to show that
the population mean is greater than 100.
•
•
•
Let’s write down the hypotheses?
Sample mean is used to estimate population
mean. Sample means vary from sample to
sample (described by sampling distribution)
What values of the sample means support
the alternative hypothesis?
When should we reject the null hypothesis?
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If we want to test population mean is
• greater than 100 (right-tailed)
• smaller than 100 (left-tailed)
• different from 100 (two-tailed)
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A problem
•
One-tailed vs. Two-tailed
tests
Rejection region (right-tailed test)
H0:  = a
Ha:  > a
Exercise: let’s draw the rejection region.
Remember that a equals total Area of Rejection
Region. Choose a small value of a as an
example.
You should now be convinced why the test is
called right-tailed.
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One population tests
Test statistic
•
Now we select a random sample from
population and calculate the sample mean.
The calculated sample mean becomes our
test statistic, as it contains evidence from the
selected sample (evidence can be against or
not against Ho)
If test statistic falls in rejection region, we
reject Ho. Otherwise, we do not reject Ho.
•
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One
population
Mean
 Z Test
1.Assumptions


Population Standard Deviation Is Known
Population Is Normally Distributed
If Population Is Not Normal, Large Sample
Size (so that the CLT holds) Is Required (In
This Case, The Sampling Distribution of
Sample Mean Will Be Approximately Normal)
2. Alternative Hypothesis Has < or > Sign
3. Z-Test Statistic
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One-Tailed Z Test
for Mean Hypotheses
H0:=a Ha:  < a
H0:=a Ha: > a
Reject H0
Reject H0
a
a
a
X
Must be significantly
below 
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One-Tailed Z Test
Finding Critical Z
To make rejection decision, we need to find out
critical value. Converting from
to Z makes this
process easier.
What Is Za given a = .025?
a = .025

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Z Test
T Test
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One-Tailed Z Test
for Mean ( Known)

Proportion
a
Small values give no
evidence against H0 in
favor of Ha. Don’t
reject!
Right-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 15
grams. Test at the .05
level. Assume normal
8 - 30 population.
X
368 gm.
Right-Tailed Z Test
Solution
H0:  = 368
Ha:  > 368
a = .05
n = 25
Critical Value(s):
Test Statistic:
Decision:
Do not reject at a = .05
Left-tailed Z test
Suppose we want to test population
mean < 100 using Z test with significance
level of 0.01. The calculated Z test
statistic is -2.5. What is the rejection
region? What is the decision?
Conclusion:
No evidence average
is more than 368
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Two-Tailed Z Test
for Mean ( Known)
Writing conclusions
1.Assumptions
2 cases:
•Reject Ho:
•Do not reject Ho:



Population Standard Deviation Is Known
Population Is Normally Distributed
If Population Is Not Normal, Large Sample
Size (n  30) Is Required (In This Case,
The Sampling Distribution of
Will Be
Approximately Normal)
2. Alternative Hypothesis Has  Sign
3. Z-Test Statistic:
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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
15 grams. Test at the
.05 level. Assume
population.
8 normal
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Two-Tailed Z Test
Solution
H0:  = 368
Ha:   368
a  .05
n  25
Critical Value(s):
Test Statistic:
Decision:
Do not reject at a = .05
Conclusion:
No evidence
average is not 368
368 gm.
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Hypothesis testing
common pitfalls
Conclusion
Choose or change hypotheses after looking at
data
1. Distinguish Types of Hypotheses

Choose or change level of significance after
looking at data
2. Describe the basics of hypothesis
testing

Do not reject Ho  accept Ho without
considering power (1-)
3. Explain the z-test for mean

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