copyright 1998, triola, elementary statistics by addison wesley longman 1 testing a claim about a...
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Copyright© 1998, Triola, Elementary Statistics by Addison Wesley Longman 3 Three Methods Discussed 1) Traditional method 2) P-value method 3) Confidence intervals Note: These three methods are equivalent, I.e., they will provide the same conclusions.TRANSCRIPT
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
1
Testing a Claim about Testing a Claim about a Mean: Large a Mean: Large
SamplesSamplesSection 7-3 Section 7-3
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
2
Assumptions 1) Sample is large (n > 30)
a) Central limit theorem applies
b) Can use normal distribution
2) Can use sample standard deviation s as estimate for if is unknown
For testing a claim about the mean of a single population
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
3
Three Methods Discussed
1) Traditional method
2) P-value method
3) Confidence intervals
Note: These three methods are equivalent, I.e., they will provide the same conclusions.
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
4
Traditional (or Classical) Method of Testing Hypotheses
GoalIdentify: whether a sample result that is
significantly different from the claimed value
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Procedure1. Identify the specific claim or hypothesis to be tested, and
put it in symbolic form.
2. Give the symbolic form that must be true when the original claim is false.
3. Of the two symbolic expressions obtained so far, put the one you plan to reject in the null hypothesis H0 (make the formula with equality). H1 is the other statement.
Or, One simplified rule suggested in the textbook: let null hypothesis H0 be the one that contains the condition of equality. H1 is the other statement.
Figure 7-4
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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4. Select the significant level based on the seriousness of a type I error. Make small if the consequences of rejecting a true H0 are severe. The values of 0.05 and 0.01 are very common.
5. Identify the statistic that is relevant to this test and its sampling distribution.
6. Determine the test statistic, the critical values, and the critical region. Draw a graph and include the test statistic, critical value(s), and critical region.
7. Reject H0 if the test statistic is in the critical region. Fail to reject H0 if the test statistic is not in the critical region.
8. Restate this previous decision in simple non-technical terms. (See Figure 7-2)
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Original claim is H0
FIGURE 7-2 Wording of 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
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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The traditional (or classical) method of hypothesis testing is actually comparing the sample test statistic value with the critical region value.
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Decision Criterion (Step 7)
Reject the null hypothesis if the test statistic is in the critical region
Fail to reject the null hypothesis if the test statistic is not in the critical region
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Fail to reject H0 Reject H0
The traditional (or classical) method of hypothesistesting is actually comparing the sample test
statistic value with the critical region value.
Fail to reject H0Reject H0
zCV
REJECT H0
zCV zTSzTS
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Fail to reject H0 Reject H0
The traditional (or classical) method of hypothesistesting is actually comparing the sample test statistic
value with the critical region value.Fail to reject H0Reject H0
Fail to reject H0 Reject H0 Fail to reject H0Reject H0
FAIL TO REJECT H0
zTS zTS
zCV
zCVzCV
REJECT H0
zCV zTSzTS
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Test Statistic for Claims about µ when n > 30
z = nx – µx
(Step 6)
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Statisticsby Addison Wesley Longman
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ExampleConjecture: “the average starting salary for a computer science gradate is $30,000 per year”.
For a randomly picked group of 36 computer science graduates, their average starting salary is $36,100 and the sample standard deviation is $8,000.
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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ExampleSolution
Step 1: µ = 30k
Step 2: µ > 30k (if believe to be no less than 30k)
Step 4: Select = 0.05 (significance level)
Step 5: The sample mean is relevant to this test and its sampling distribution is approximately normal (n = 36 large, by CLT)
Step 3: H0: µ = 30k versus H1: µ > 30k
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Central Limit Theorem:Assume the conjecture is true!
z = x – µx
nTest Statistic:
Critical value = 1.64 * 8000/6 + 30000 = 32186.67
30 K( z = 0)
Fail to reject H0 Reject H0
32.2 k( z = 1.64 )
(Step 6)
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Central Limit Theorem:Assume the conjecture is true!
z = x – µx
nTest Statistic:
Critical value = 1.64 * 8000/6 + 30000 = 32186.67
30 K( z = 0)
Fail to reject H0 Reject H0
32.2 k( z = 1.64 )
Sample data: z = 4.575
x = 36.1k or
(Step 7)
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Example
Conclusion: Based on the sample set, there is sufficient evidence to warrant rejection of the claim that “the average starting salary for a computer science gradate is $30,000 per year”.
Step 8:
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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P-Value Methodof Testing Hypotheses
• DefinitionP-Value (or probability value)
the probability of getting a value of the sample test statistic that is at least as extreme as the one found from the sample data, assuming that the null hypothesis is true• Measures how confident we are in rejecting the null
hypothesis
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Procedure is the same except for steps 6 and 7Step 6: Find the P-value
Step 7: Report the P-value
Reject the null hypothesis if the P-value is less than or equal to the significance level
Fail to reject the null hypothesis if the P-value is greater than the significance level
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Highly statistically significant
Very strong evidence against the null hypothesis
Statistically significant
Adequate evidence against the null hypothesis
Insufficient evidence against the null hypothesis
Less than 0.01
P-value Interpretation
Greater than 0.05
0.01 to 0.05
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Figure 7-7 Finding P-Values
Isthe test statistic
to the right or left ofcenter
?
P-value = areato the left of the test statistic
P-value = twice the area to the left of the test statistic
P-value = areato the right of the test statistic
Left-tailed Right-tailed
RightLeft
Two-tailed
P-value = twice the area to the right of the test statistic
Whattype of test
?
Start
µ µ µ µ
P-value P-value is twicethis area
P-value is twicethis area
P-value
Test statistic Test statistic Test statistic Test statistic
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
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Central Limit Theorem:Assume the conjecture is true!
z = x – µx
nTest Statistic:
30 K 36.1 k
(Step 6)
P-value = areato the right of the test statistic
Z = 36.1 - 308 / 6 = 4.575 P-value = .0000024
Copyright© 1998, Triola, Elementary
Statisticsby Addison Wesley Longman
23
Central Limit Theorem:
30 K 36.1 k
(Step 7)
P-value = areato the right of the test statistic
Z = 36.1 - 308 / 6 = 4.575 P-value = .0000024
P-value < 0.01
Highly statistically significant (Very strong evidence against the null hypothesis)