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Week 8

Fundamentals of Hypothesis Testing: One-Sample Tests

Statistics

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Goals of this note

After completing this noe, you should be able to:

Formulate null and alternative hypotheses for applications involving a single population mean or proportion

Formulate a decision rule for testing a hypothesis

Know how to use the p-value approaches to test the null hypothesis for both mean and proportion problems

Know what Type I and Type II errors are

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What is a Hypothesis?

A hypothesis is a claim (assumption) about a population parameter:

population mean

population proportion

The average number of TV sets in U.S. homes is equal to three ( )

A marketing company claims that it receives 8% responses from its mailing. ( p=.08 )

3μ

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States the assumption to be tested

Example: The average number of TV sets in U.S. Homes is equal to three ( )

Is always about a population parameter, not about a sample statistic

The Null Hypothesis, H0

3μ:H0

3μ:H0 3X:H0

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The Null Hypothesis, H0

Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until

proven guilty Refers to the status quo Always contains “=” , “≤” or “” sign May or may not be rejected

(continued)

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The Alternative Hypothesis, H1

Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S.

homes is not equal to 3 ( H1: μ ≠ 3 )

Challenges the status quo Never contains the “=” , “≤” or “” sign Is generally the hypothesis that is believed (or

needs to be supported) by the researcher

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Hypothesis Testing

We assume the null hypothesis is true If the null hypothesis is rejected we have

proven the alternate hypothesis If the null hypothesis is not rejected we have

proven nothing as the sample size may have been to small

Population

Claim: thepopulationmean age is 50.(Null Hypothesis:

REJECT

Supposethe samplemean age is 20: X = 20

SampleNull Hypothesis

20 likely if μ = 50?Is

Hypothesis Testing Process

If not likely,

Now select a random sample

H0: μ = 50 )

X

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Do not reject H0 Reject H0Reject H0

There are two cutoff values (critical values), defining the regions of rejection

Sampling Distribution of

/2

0

H0: μ = 50

H1: μ 50

/2

Lower critical value

Upper critical value

50 X

X

20 Likely Sample Results

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Level of Significance,

Defines the unlikely values of the sample statistic if the null hypothesis is true Defines rejection region of the sampling distribution

Is designated by , (level of significance)

Typical values are .01, .05, or .10 Is the compliment of the confidence coefficient Is selected by the researcher before sampling

Provides the critical value of the test

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Level of Significance and the Rejection Region

H0: μ ≥ 3

H1: μ < 30

H0: μ ≤ 3

H1: μ > 3

Represents critical value

Lower tail test

Level of significance =

0Upper tail test

Two tailed test

Rejection region is shaded

/2

0

/2H0: μ = 3

H1: μ ≠ 3

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Type I Error When a true null hypothesis is rejected The probability of a Type I Error is

Called level of significance of the test Set by researcher in advance

Type II Error Failure to reject a false null hypothesis The probability of a Type II Error is β

Errors in Making Decisions

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Example

The Truth

Verdict

Innocent No error Type II Error

Guilty Type I Error

Possible Jury Trial Outcomes

Guilty Innocent

No Error

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Outcomes and Probabilities

Actual SituationDecision

Do NotReject

H0

No error (1 - )

Type II Error ( β )

RejectH0

Type I Error( )

Possible Hypothesis Test Outcomes

H0 False H0 TrueKey:

Outcome(Probability)

No Error ( 1 - β )

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Type I & II Error Relationship

Type I and Type II errors can not happen at the same time

Type I error can only occur if H0 is true

Type II error can only occur if H0 is false

If Type I error probability ( ) , then

Type II error probability ( β )

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p-Value Approach to Testing

p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H0 is

true

Also called observed level of significance

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p-Value Approach to Testing

Convert Sample Statistic (e.g. ) to Test Statistic (e.g. t statistic )

Obtain the p-value from a table or computer

Compare the p-value with

If p-value < , reject H0

If p-value , do not reject H0

X

(continued)

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8 Steps in Hypothesis Testing

1. State the null hypothesis, H0

State the alternative hypotheses, H1

2. Choose the level of significance, α

3. Choose the sample size, n

4. Determine the appropriate test statistic to use

5. Collect the data

6. Compute the p-value for the test statistic from the sample result

7. Make the statistical decision: Reject H0 if the p-value is less than alpha

8. Express the conclusion in the context of the problem

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Hypothesis Tests for the Mean

Known Unknown

Hypothesis Tests for

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Hypothesis Testing Example

Test the claim that the true mean # of TV sets in U.S. homes is equal to 3.

1. State the appropriate null and alternative hypotheses

H0: μ = 3 H1: μ ≠ 3 (This is a two tailed

test) 2. Specify the desired level of significance

Suppose that = .05 is chosen for this test 3. Choose a sample size

Suppose a sample of size n = 100 is selected

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4. Determine the appropriate Test

σ is unknown so this is a t test 5. Collect the data

Suppose the sample results are

n = 100, = 2.84 s = 0.8 6. So the test statistic is:

The p value for n=100, =.05, t=-2 is .048

2.0.08

.16

100

0.832.84

n

sμX

t

Hypothesis Testing Example(continued)

X

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7. Is the test statistic in the rejection region?

Reject H0 if p is < alpha; otherwise do not reject H0

Hypothesis Testing Example(continued)

The p-value .048 is < alpha .05, we reject the null hypothesis

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8. Express the conclusion in the context of the problem

Since The p-value .048 is < alpha .05, we have rejected the null hypothesis

Thereby proving the alternate hypothesis

Conclusion: There is sufficient evidence that the mean number of TVs in U.S. homes is not equal to 3

Hypothesis Testing Example(continued)

If we had failed to reject the null hypothesis the conclusion would have been: There is not sufficient evidence to reject the claim that the mean number of TVs in U.S. home is 3

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One Tail Tests

In many cases, the alternative hypothesis focuses on a particular direction

H0: μ ≥ 3

H1: μ < 3

H0: μ ≤ 3

H1: μ > 3

This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3

This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3

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Reject H0 Do not reject H0

There is only one

critical value, since

the rejection area is

in only one tail

Lower Tail Tests

-t 3

H0: μ ≥ 3

H1: μ < 3

Critical value

X

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Reject H0Do not reject H0

Upper Tail Tests

tα3

H0: μ ≤ 3

H1: μ > 3

There is only one

critical value, since

the rejection area is

in only one tail

Critical value

t

X

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Assumptions of the One-Sample t Test

The data is randomly selected

The population is normally distributed orthe sample size is over 30 and the population is not highly skewed

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Hypothesis Tests for Proportions

Involves categorical values

Two possible outcomes “Success” (possesses a certain characteristic)

“Failure” (does not possesses that characteristic)

Fraction or proportion of the population in the “success” category is denoted by p

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Proportions

Sample proportion in the success category is denoted by ps

When both np and n(1-p) are at least 5, ps can be approximated by a normal distribution with mean and standard deviation

sizesample

sampleinsuccessesofnumber

n

Xps

pμ sp n

p)p(1σ

sp

(continued)

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The sampling distribution of ps is approximately normal, so the test statistic is a Z value:

Hypothesis Tests for Proportions

n)p(p

ppZ

s

1

np 5and

n(1-p) 5

Hypothesis Tests for p

np < 5or

n(1-p) < 5

Not discussed in this chapter

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Example: Z Test for Proportion

A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the = .05 significance level.

Check:

n p = (500)(.08) = 40

n(1-p) = (500)(.92) = 460

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Z Test for Proportion: Solution

= .05

n = 500, ps = .05p-value for -2.47 is .0134Decision:Reject H0 at = .05

H0: p = .08

H1: p

.08

Critical Values: ± 1.96

Test Statistic:

Conclusion:z0

Reject Reject

.025.025

1.96

-2.47

There is sufficient evidence to reject the company’s claim of 8% response rate.

2.47

500.08).08(1

.08.05

np)p(1

ppZ

s

-1.96

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Potential Pitfalls and Ethical Considerations

Use randomly collected data to reduce selection biases Do not use human subjects without informed consent Choose the level of significance, α, before data

collection Do not employ “data snooping” to choose between one-

tail and two-tail test, or to determine the level of significance

Do not practice “data cleansing” to hide observations that do not support a stated hypothesis

Report all pertinent findings

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Summary

Addressed hypothesis testing methodology

Discussed critical value and p–value approaches to hypothesis testing

Discussed type 1 and Type2 errors

Performed two tailed t test for the mean (σ unknown)

Performed Z test for the proportion

Discussed one-tail and two-tail tests

Addressed pitfalls and ethical issues