Hypothesis Testing
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Transcript of Hypothesis Testing
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Hypothesis Testing
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Hypothesis is a claim or statement about a property of a population.
Hypothesis Testing is to test the claim or statement
Example: A conjecture is made that “the average starting salary for computer science gradate is Rs 45,000 per month”.
Hypothesis Testing
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Hypothesis Testing Null Hypothesis (H 0): is the
statement being tested in a test of hypothesis.
Alternative Hypothesis (H 1): is
what is believe to be true if the null hypothesis is false.
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Null Hypothesis
Must contain condition of equality
=, ³, or £ Test the Null Hypothesis directly
Reject H 0 or fail to reject H 0
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Alternative Hypothesis Must be true if H0 is false ¹, <, > ‘opposite’ of Null HypothesisExample H0 : µ = 30 versus H1 : µ > 30 or H1 : µ < 30
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State the Null Hypothesis (H0: m ³ 3)
State its opposite, the Alternative
Hypothesis (H1: m < 3)
Hypotheses are mutually exclusive
Identify the Problem
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Population
Assume thepopulationmean age is 50.(Null Hypothesis)
REJECT
The SampleMean Is 20
SampleNull Hypothesis
50?20 mXIs
Hypothesis Testing Process
No, not likely!
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Sample Meanm = 50
Sampling DistributionIt is unlikely that we would get a sample mean of this value ...
... if in fact this were the population mean.
... Therefore, we reject the null
hypothesis that m = 50.
20H0
Reason for Rejecting H0
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• Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling Distribution
• Designated a (alpha) Typical values are 0.01, 0.05, 0.10
• Selected by the Researcher at the Start• Provides the Critical Value(s) of the Test
Level of Significance, a
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Level of Significance and the Rejection Region
H0: m ³ 3 H1: m < 3
0
0
0
H0: m £ 3 H1: m > 3
H0: m 3 H1: m ¹ 3
a
a
a/2
Critical Value(s)
Rejection Regions
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Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error is a
Called Level of Significance Type II Error
Do Not Reject False Null Hypothesis Probability of Type II Error Is b (Beta)
Errors in Making Decisions
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H0: InnocentJury Trial Hypothesis Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct ErrorDo NotReject
H0
1 - a Type IIError (b )
Guilty Error Correct RejectH0
Type IError( a ) (1 - b)
Result Possibilities
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Type I Error The mistake of rejecting the null hypothesis
when it is true.
The probability of doing this is called the significance level, denoted by a (alpha).
Common choices for a: 0.05 and 0.01
Example: rejecting a perfectly good parachute and refusing to jump
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Type II Error
The mistake of failing to reject the null hypothesis when it is false.
Denoted by ß (beta)
Example: Failing to reject a defectiveparachute and jumping out of a
plane with it.
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Critical Region
Set of all values of the test statistic that would cause a rejection of the null hypothesis.
CriticalRegion
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Critical Region
Set of all values of the test statistic that would cause a rejection of the null hypothesis.
CriticalRegion
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Critical Region
Set of all values of the test statistic that would cause a rejection of the null hypothesis.
CriticalRegions
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Critical ValueValue (s) that separates the critical region from the values that would not lead to a rejection of H 0.
Critical Value( z score )
Fail to reject H0Reject H0
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Original claim is H0
Conclusions in Hypothesis Tests
Doyou reject
H0?.
Yes
(Reject H0)
“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“The sample datasupports the claim that . . . (original claim).”
“There is not sufficientevidence to support the claim that. . . (original claim).”
Doyou reject
H0?
Yes
(Reject H0)
No(Fail toreject H0)
No(Fail toreject H0)
(This is theonly case inwhich theoriginal claimis rejected).
(This is theonly case inwhich theoriginal claimis supported).
Original claim is H1
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Left-tailed TestH0: µ ³ 200
H1: µ < 200
200
Values that differ significantly
from 200
Fail to reject H0Reject H0
Points Left
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Right-tailed TestH0: µ £ 200
H1: µ > 200
Values that differ significantly
from 200200
Fail to reject H0 Reject H0
Points Right
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Two-tailed TestH0: µ = 200H1: µ ¹ 200
Means less than or greater than
Fail to reject H0Reject H0 Reject H0
200
Values that differ significantly from 200
a is divided equally between the two tails of the critical
region
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DefinitionTest Statistic: is a sample statistic or value based on sample data Example:
z = x – µxs / n
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Question: How can we justify/test this conjecture?
A. What do we need to know to justify this conjecture?
B. Based on what we know, how should we justify this conjecture?
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Answer to A: Randomly select, say 100, computer
science graduates and find out their annual salaries
---- We need to have some sample observations, i.e., a sample set!
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Answer to B: That is what we will learn in this chapter ---- Make conclusions based on the sample
observations
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Statistical Reasoning Analyze the sample set in an attempt to
distinguish between results that can easily occur and results that are highly unlikely.
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Figure 7-1 Central Limit Theorem:
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Figure 7-1 Central Limit Theorem:Distribution of Sample Means
µx = 30k
Likely sample means
Assume the conjecture is true!
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Figure 7-1 Central Limit Theorem:Distribution of Sample Means
z = –1.96
x = 20.2kor
z = 1.96
x = 39.8kor
µx = 30k
Likely sample means
Assume the conjecture is true!
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Figure 7-1 Central Limit Theorem:Distribution of Sample Means
z = –1.96
x = 20.2kor
z = 1.96
x = 39.8kor
Sample data: z = 2.62
x = 43.1k or
µx = 30k
Likely sample meansAssume the conjecture is true!
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COMPONENTS OF AFORMAL HYPOTHESIS TEST
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TWO-TAILED,LEFT-TAILED,RIGHT-TAILEDTESTS
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Problem 1Test µ = 0 against µ > 0, assuming
normally and using the sample [multiples of 0.01 radians in some revolution of a satellite]
1, -1, 1, 3, -8, 6, 0 (deviations of azimuth)
Choose α = 5%.
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Problem 2
In one of his classical experiments Buffon
obtained 2048 heads in tossing a coin 4000
times. Was the coin fair?
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Problem 3
In one of his classical experiments K
Pearson obtained 6019 heads in 12000
trials. Was the coin fair?
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Problem 5
Assuming normality and known variance
б2 = 4, test the hypotheses µ = 30 using a
sample of size 4 with mean X = 28.5 and
choosing α = 5%.
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Problem 7
Assuming normality and known variance
б2 = 4, test the hypotheses µ = 30 using a
sample of size 10 with mean X = 28.5. What
is the rejection region in case of a two sided
test with α = 5%.
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Problem 9A firm sells oil in cans containing 1000 g oil per can
and is interested to know whether the mean weight
differs significantly from 1000 g at the 5% level, in
which case the filling machine has to be adjusted. Set
up a hypotheses and an alternative and perform the
test, assuming normality and using a sample of 20
fillings with mean 996 g and Standard Deviation 5g.
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Problem 11
If simultaneous measurements of electric voltage
by two different types of voltmeter yield the
differences (in volts)
0.8, 0.2, -0.3, 0.1, 0.0, 0.5, 0.7 and 0.2
Can we assert at the 5% level that there is no
significant difference in the calibration of the two
types of instruments? Assume normality.