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THE BIG PICTURE HANDOUT

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

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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