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The future is a vain hope, the past is a distracting thought. Uphold our loving

kindness at this instant, and be committed to our duties and responsibilities right now.

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Applied Statistics Using SAS and SPSS

Topic: Hypothesis Testing

By Prof Kelly Fan, Cal State Univ, East Bay

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Hypothesis Testing

A statistical hypothesis is an assertion or conjecture concerning one or more populations.

Agenda:

1.Types of tests

2.Types of errors

3.P-value

4.Summary of tests

5.Assumption checking

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Types of Tests

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Types of Tests

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Types of Tests

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Good!Good!(Correct!)(Correct!)

H0 true H0 false

Type II Type II Error, or Error, or ““ Error” Error”

Type I Type I Error, or Error, or ““ Error” Error”

Good!Good!(Correct)(Correct)

we accept H0

we reject H0

Types of Errors

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= Probability of Type I error = P(rej. H0|H0 true)

= Probability of Type II error= P(acc. H0|H0 false)

We often preset called significance level. The value of depends on the specifics of the H1 (and most often in the real world, we don’t know these specifics).

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C=141

EXAMPLE: H0 : < 100

H1 : >100

Suppose the Critical Value = 141:

=100

X

12 = P (X < 141|H0 false)

141What is ?

= P (X < 141/= 150)= .3594.3594

= 150

= 150

141

= P (X < 141/= 160)= .2236.2236

= 160

= 160

141

= P (X < 141/= 170)= .1230.1230

= 170

= 170

141

= P (X < 141/= 180)

= 180

= 180

These are values

corresp.to a value of 25 for the Std. Dev. of X

= .0594= .0594

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Note: Had been preset at .025 (instead of .05), C would have been 149 (and would be larger); had been preset at .10, C would have been 132 and would be smaller.

and “trade off”.

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P Value

Idea: The largest “risk” we pay to reject H0(the observed type I error rate)(or the observed significance level)

When will we reject Ho ? What is the formula to calculate the largest risk?

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Steps of Hypothesis Tests

1. Set up Ho and Ha properly

2. Preset level (the significant level)

3. Select an appropriate test

4. Calculate its p-value

5. Reject Ho if p-value < or = the significant level; otherwise fail to reject Ho

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Set Up Hypothesis Properly

Conjecture: The fraction of defective product in a certain process is at most 10%.

Which error is more seriously? Incorrectly claim this conjecture is true? false?

The “=“ sign must be in Ho

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One Population

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Two Populations

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Assumption Checking

1. Tests/graphs for normality

2. Tests for equal variances

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Example: Mortar Strength

The tension bond strength of cement mortar is an important characteristic of the product. An engineer is interested in comparing the strength of a modified formulation in which polymer latex emulsions have been added during mixing to the strength of the unmodified mortar. The experimenter has collected 10 observations on strength for the modified formulation and another 10 observations for the unmodified formulation.

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Example: Mortar Strength

ModifiedUnmodified

16.85 17.5016.40 17.6317.21 18.2516.35 18.0016.52 17.8617.04 17.7516.96 18.2217.15 17.9016.59 17.9616.57 18.15

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SAS/SPSS Data Input

SPSS: One variable one column in the work sheet

SAS: One variable one name

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Normality Tests

Tests for Normality

Test --Statistic--- -----p Value------

Shapiro-Wilk W 0.918255 Pr < W 0.0917Kolmogorov-Smirnov D 0.134926 Pr > D >0.1500Cramer-von Mises W-Sq 0.081542 Pr > W-Sq 0.1936Anderson-Darling A-Sq 0.537514 Pr > A-Sq 0.1503

SAS: PROC UNIVARIATE DATA=** NORMAL PLOT;

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Normality Tests

SPSS: Analyze >> Descriptive Statistics >> Explore >> Plots , Normality plots with tests

Tests of Normality

.135 20 .200* .918 20 .092strengthStatistic df Sig. Statistic df Sig.

Kolmogorov-Smirnova

Shapiro-Wilk

This is a lower bound of the true significance.*.

Lilliefors Significance Correctiona.

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Two-sample t Tests and Equal-variance Tests

SAS: PROC TTEST DATA=** ;

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SPSS: equal-variance tests: Homework for ST3900 studentsSPSS: two-sample t tests as below

Two-sample t Tests and Equal-variance Tests

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Research Question

A researcher claims that a new series of math courses for elementary school is more effective than the current one. Two (1st grade) classes of students are selected to perform an experiment to verify this claim. How would you conduct the experiment to avoid confounding variables as much as possible?

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Paired Samples

If the same set of sources are used to obtain data representing two populations, the two samples are called paired. The data might be paired: As a result of the data from certain “before” and

“after” studies From matching two subjects to form “matched

pairs”

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Tests for Paired Samples

Calculate the pair differencesProceed as in one sample case

Notes: SAS: all variables must be included in data SPSS: create/calculate all variables we need

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Review: Confidence Interval

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One Population

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Two Populations