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9.1 PART ONE: General Information about Hypothesis Testing

Writing the claim, Ho, and H1.

-Identify the claim stated in the problem and express it in symbolic form.

-Give the symbolic form of the claim that must be false if the claim is true.

-Of the two symbolic expressions obtained so far, let the alternative hypothesis

H1 be the one not containing the equality so H1 uses the symbol or < or >.

-Let the null hypothesis Ho be the symbolic expression that the parameter equals

the fixed value being considered.

Example) Test the claim that the mean is greater than 4.

Claim: >4 BUT 4 will be false when the claim is true so we choose the

statement with or < or > to be H1 and Ho always has the = sign.)

Ho: =4

H1: >4

Example) Test the claim that the mean is equal to 10.

Claim: =10 BUT 10 will be false when the claim is true so we choose the

statement with or < or > to be H1 and Ho always has the = sign.)

Ho: =10

H1: 10

Example) Test the claim that the mean is less than or equal to 5.

Claim: 5 BUT >5 will be false when the claim is true so we choose the

statement with or < or > to be H1 and Ho always has the = sign.)

Ho: =5

H1: > 5

Example) Test the claim that the mean is not equal to 7.

Claim: 7 BUT =7 will be false when the claim is true so we choose the

statement with or < or > to be H1 and Ho always has the = sign.)

Ho: =7

H1: 7

≠

µ µ ≤

≠

µ

µ

µ µ ≠

≠

µ

µ ≠

µ ≤ µ

≠

µ

µ

µ ≠ µ

≠

µ

µ ≠

2

Symbols

Mean Standard

Deviation

Variance Proportion Correlation

Coefficient

Slop

e

Population

Parameters

“mu”

“sigma”

“sigma

squared”

P

“rho”

“beta

sub

1”

Sample

Statistics

“xbar”

s

“P hat”

r m

Note: The population parameters are used in the claim, Ho, and H1.

Rules for the p value method: for DECISION about Ho

If , then reject Ho.

If , then we do not reject Ho.

Note: We assume Ho is true from the start. If based on the sample data, Ho is

unusual, then we reject Ho. “Unusual” is less than or equal to the alpha level. For

example) = .05 or 5%. In this case we assume Ho is true unless the probability of

Ho occurring based on the sample data, p, is unusual which is 5% or less.

Case 1: = .05 or 5% and p = .03 or 3% so , so we reject Ho.

Case 2: = .05 or 5% and p = .10 or 10% so , then we do not reject Ho.

To write the LONG CONCLUSION about the CLAIM

Decision Claim

Claim has “condition of

equality” means Claim

has or =

Claim does not have

“condition of equality”

means Claim has or

< or >

Reject Ho There is enough evidence to

reject the claim.

There is enough evidence to

support the claim.

Do not Reject Ho There is not enough evidence

to reject the claim.

There is not enough evidence

to support the claim.

Hypothesis testing is like the courtroom…. we always assume innocence unless

proven guilty. We always assume Ho is true unless there is enough evidence to

show otherwise. That is why when the claim has the condition of equality which is

similar to Ho, we use the word REJECT and when the claim does not have the

condition of equality which is unlike Ho, we use the word SUPPORT.

µ σ σ

2 ρ β1

x s

2

P̂

p ≤ α

p > α

α

α p ≤ α

α p > α

≥ or ≤ ≠

3

CONCLUSION about Ho or H1

If there is sufficient evidence to reject Ho, then we conclude that H1 is true.

If there is not enough evidence to reject Ho, then we conclude that Ho might

be true, but we never conclude that Ho is true. Recall in the courtroom, the

verdict is man is guilty or man is not guilty, but never man is innocent.

Note: When we do not reject Ho, it just means that the evidence was not strong

enough to reject Ho.

Interpreting Confidence Intervals

Ex) A 95% confidence interval with n = 200, for the difference between two

means, has already been calculated to be:

(.381, .497).

Correct Interpretation

“We are 95% confident that the true value of lies in the confidence

interval.”

This means if we were to select many different samples (for example 100 samples)

of size 200 and construct the corresponding confidence intervals, 95% of them (95)

would actually contain the true value of and 5% (5) of them would not.

Note: In this correct interpretation, the level 95% refers to the success rate of the

process being used to estimate , and it does not refer to the difference of the

population means.

Incorrect Interpretation

There is a 95% chance that the true value of lies in the confidence interval.

Note: Where does come from? It comes from rewriting Ho.

Ho:

µ1 − µ2

µ1 − µ2

µ1 − µ2

µ1 − µ2

µ1 − µ2

µ1 − µ2

µ1 = µ2 or µ1 − µ2 = 0

4

PART TWO: One Sample & Two Sample Tests Conditions

How to know when to use which test?

I. ONE-SAMPLE TESTS:

MEAN

Z test for a Mean

is known and the population is normal OR for any population when

is known and n 30.

T test for a Mean

is not known and the population is normal OR for any population

when is not known and n 30.

PROPORTION 1-Prop Z test

II. TWO-SAMPLE TESTS

TWO MEANS

2 Sample Z-test

Independent Samples. Both populations normal OR both sample sizes

are at least 30. known.

2 Sample T-test

Independent Samples. Both populations normal OR both sample sizes

are at least 30. not known.

T-Test for dependent samples

Dependent samples. Both populations normal OR both sample sizes are

at least 30.

TWO PROPORTIONS 2-Prop Z Test

TWO STANDARD DEVIATIONS 2 Sample F Test

Table below summarizes parametric and nonparametric tests (do

not require any specific conditions concerning the shapes of

population distributions or the values of population parameters)

Always use the parametric test if the conditions are satisfied.

σ σ

≥

σ

σ ≥

σ1 and σ 2

σ1 and σ 2

5

TEST APPLICATION Parametric test ch 9-14

Nonparametric test Ch15

One -sample tests Z test for Mean Sign test for Median

T test for Mean

Two-sample tests

Dependent T test Sign test/Wilcoxon

Signed Rank

Independent 2 sample z test Wilcoxon Rank Sum

2 sample t test

3 or more samples One-Way ANOVA Kruskal- Wallis Test

Correlation Pearson Correlation

Coefficient

Spearman rank

correlation coefficient

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