Psyc 235:Introduction to Statistics

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http://www.psych.uiuc.edu/~jrfinley/p235/

Stuff

• Thursday: office hours hands-on help with specific problems

• Next week labs: demonstrations of solving various types

of hypothesis testing problems

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X

Population

Sample

SamplingDistribution

€

X

size = n

(of the mean)

Descriptive vs Inferential

• Descriptivedescribe the data you’ve got if those data are all you’re interested in,

you’re done.

• Inferentialmake inferences about population(s) of

values (when you don’t/can’t have complete

data)

Inferential

• Point Estimate• Confidence Interval• Hypothesis Testing

1 population parameter z, t tests

2 pop. parameters z, t tests on differences

3 or more?... ANOVA!

Hypothesis Testing

1. Choose pop. parameter of interest• (ex: )

2. Formulate null & alternative hypotheses

• assume the null hyp. is true

3. Select test statistic (e.g., z, t) & form of sampling distribution

• based on what’s known about the pop., & sample size

Defining our hypothesis

• H0= the Null hypothesisUsually designed to be the situation of

no difference The hypothesis we test.

• H1= the alternative hypothesisUsually the research related hypothesis

Null Hypothesis(~Status Quo)

Examples:

• Average entering age is 28 (until shown different)

• New product no different from old one (until shown better)

• Experimental group is no different from control

group (until shown different)

• The accused is innocent (until shown guilty)

€

Null Hypothesis Alternative Hypothesis

H0 H1,HA ,Ha

H0 : μ = μ0 Ha : μ ≠ μ0

H0 : μ ≥ μ0 (μ = μ0) Ha : μ < μ0

H0 : μ ≤ μ0 (μ = μ0) Ha : μ > μ0

- Ha is the hypothesis you are gathering evidence in support of.

- H0 is the fallback option = the hypothesis you would like to reject.

- Reject H0 only when there is lots of evidence against it.

- A technicality: always include “=” in H0

- H0 (with = sign) is assumed in all mathematical calculations!!!

Decision Tree for Hypothesis Testing

PopulationStandard Deviationknown?

Yes

No

Pop. Distributionnormal?

n large?(CLT)

Yes

No Yes

No

Yes

No YesNo

z-score

z-score

Can’t do it

Can’t do it

t-score

t-score

Test stat.

Standard normaldistribution

t distribution

Selecting a distribution

Hypothesis Testing

1. Choose pop. parameter of interest• (ex: )

2. Formulate null & alternative hypotheses

• assume the null hyp. is true

3. Select test statistic (e.g., z, t) & form of sampling distribution

• based on what’s known about the pop., & sample size

Hypothesis Testing

4. Calculate test stat.:

5. Note: The null hypothesis implies a certain sampling distribution

6. if test stat. is really unlikely under Ho, then reject Ho

• HOW unlikely does it need to be? determined by

€

sample stat.− pop. param. under H0

std. dev, of sampling distribution

Three equivalent methods of hypothesis testing

(=significance level)

0

0

0

HX

H

H

reject then, value-p If . observed of value-p Compute

reject theninterval, confidence innot If interval. confidence -1 Compute

reject then value,critical thanextreme more statistic edstandardiz If

statistic. edstandardiz Compute

0

<

p-value: prob of getting test stat at least as extreme if Ho really true.

Hypothesis Testing as a Decision Problem

Great!Type II Error

Great!Type IError

true0H false 0H

€

Fail to

Reject H0

Reject 0H

Level ceSignifican

PH

=)Error I (Type

of Rejection False 0

β P

H

H

= Error)II (Type

false given

retained

0

0

Power:

1 – P(Type II error)

Our ability to rejectthe null hypothesiswhen it is indeed

false

Depends on sample sizeand how much the null and alternative hypotheses differ

ERRORS

• Type I errors (): rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit.

• Type II errors (β): failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit.

Type I and Type II errors

-6 -4 -2 0 2 4 6 8 10

decisioncriterion

Power (1- β

β

ANOVA: Analysis of Variance

• a method of comparing 3 or more group means simultaneously to test whether the means of the corresponding populations are equal (why not just do a bunch of 2-sample t-

tests?...) inflation of Type I error rate

ANOVA: 1-Way

• You have sample data from several different groups

• “One-way” refers to one factor.• Factor = a categorical variable that

distinguishes the groups. • Level (group) of the factor refers to

the different values that the categorical variable can take.

ANOVA: 1-Way

• Examples of Factors & groups: Factor: Political Affiliation

groups: Democrat, Republican, Independent X=annual income

Factor: Studying Method groups: Re-read notes, practice test, do

nothing (control) X=score on exam

ANOVA: 1-Way

• So you’ve got 3(+) sets of sample data, from 3 different populations.

• You want to test whether those 3 populations all have the same mean ()

• Null Hypothesis: H0: 1=2=3 (all pop. means are same)

H1: all pop. means are NOT the same!

• [draw examples on chalkboard]

ANOVA: Assumptions

• Normality populations are normally distributed

• Homogeneity of variance populations have same variance (2)

• 1-Way “Independent Samples”: groups are independent of each other

ANOVA: the idea

• Two ways to estimate 2

MSB: Mean Square Between Group (aka MSE: MS Error) based on how spread out the sample means are from each

other. Variation Between Samples

MSW: Mean Square Within Group based on the spread of data within each group Variation within Samples

• If the 3(+) populations really do have same mean, then these 2 #s should be ~ the same

• If NOT, then MSB should be bigger.

ANOVA: calculating

• MSB: Variation between samples (sample size) * (variance of sample means)

if sample sizes are the same in all groups note: use the “sample variance” formula

• MSW: Variation within samples (mean of sample variances)

ANOVA: the F statistic

• So how to compare MSB and MSW?

• Under H0: F≈1• So calculate your F test statistic and

compare to F distribution, see if it falls in region of rejection. [chalkboard]

• note: F one-tailed!€

Fdfn,dfd =MSB

MSW

ANOVA: F & df

• F distribution requires specification of 2 degrees of freedom values

• DFn: degrees of freedom numerator: (# of groups) - 1

• DFd: degrees of freedom denominator: (total sample size (N)) - (# of groups)

ANOVA: example

• Groups: adults w/ 3 different activity levels• X=% REM sleep

• MSB=(sample size)(variance of sample means)=...• MSW=(mean of sample variance)=...• F=MSB/MSW=...• dfn=# groups - 1=... dfd=Ntotal-#groups=...• Fcritical=... p-value=...

GroupSample

sizeSample mean

Sample variance

Very Active 10 26.6 3Moderately Active 10 25.1 14.4Inactive 10 26.7 4.7