Copyright 1998, Triola, Elementary Statistics by Addison Wesley Longman 1 Testing a Claim about a...

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



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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



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.



4

Traditional (or Classical) Method of Testing Hypotheses

GoalIdentify: whether a sample result that is

significantly different from the claimed value



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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



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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)



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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



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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.



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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



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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



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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



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Test Statistic for Claims about µ when n > 30

z = nx – µx

(Step 6)



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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.



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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



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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)


32.2 k( z = 1.64 )

(Step 6)



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z = x – µx

nTest Statistic:

Critical value = 1.64 * 8000/6 + 30000 = 32186.67

30 K( z = 0)


32.2 k( z = 1.64 )

Sample data: z = 4.575

x = 36.1k or

(Step 7)



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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:



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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



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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



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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



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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



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z = x – µx

nTest Statistic:

30 K 36.1 k

(Step 6)


Z = 36.1 - 308 / 6 = 4.575 P-value = .0000024



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Central Limit Theorem:

30 K 36.1 k

(Step 7)


Z = 36.1 - 308 / 6 = 4.575 P-value = .0000024

P-value < 0.01

Highly statistically significant (Very strong evidence against the null hypothesis)

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