Testing Hypotheses I

Lesson 9

Descriptive vs. Inferential Statistics

Descriptive quantitative descriptions of

characteristics Inferential Statistics

Drawing conclusions about parameters ~

Hypothesis Testing

Hypothesis testable assumption about a parameter should conclusion be accepted? final result a decision: YES or NO qualitative not quantitative

General form of test statistic ~

Hypothesis Test: General Form

error

effect

chance todue difference

groupsbetween difference statistictest

X

obs

Xz

variationicunsystemat

variationsystematic statistictest

Evaluating Hypotheses

Hypothesis: sample comes from this population

Two Hypotheses Testable predictions Alternative Hypothesis: H1

also scientific or experimental hypothesis there is a difference between groups Or there is an effect Reflects researcher’s prediction

Null Hypothesis: H0

there is no difference between groups Or there is no effect This is hypothesis we test ~

Conclusions about Hypotheses

Cannot definitively “prove” or “disprove” Logic of science built on “disproving”

easier than “proving” State 2 mutually exclusive & exhaustive

hypotheses if one is true, other cannot be true

Testing H0

Assuming H0 is true, what is probability we would obtain these data? ~

Hypothesis Test: Outcomes

Reject Ho accept H1 as true

supported by data statistical significance

difference greater than chance Fail to reject

“Accepting” Ho data are inconclusive ~

Hypotheses & Directionality

Directionality affects decision criterion Direction of change of DV

Nondirectional hypothesis Does reading to young children affect

IQ scores? Directional hypothesis

Does reading to young children increase IQ scores? ~

Nondirectional Hypotheses

2-tailed test Similar to confidence interval Stated in terms of parameter

Hypotheses H1 : 100

Ho : = 100 Do not know what effect will be

can reject H0 if increase or decrease in IQ scores ~

Directional Hypotheses

1- tailed test predict that effect will be increase

or decrease Only predict one direction

Prediction of direction reflected in H1

H1: > 100 Ho: < 100 Can only reject H0 if change is in

same direction H1 predicts ~

Errors

“Accept” or reject Ho

only probability we made correct decision

also probability made wrong decision Type I error ()

incorrectly rejecting Ho e.g., may think a new antidepressant is

effective, when it is NOT ~

Errors Type II error ()

incorrectly “accepting” Ho e.g., may think a new antidepressant is not

effective, when it really is Do not know if we make error

Don’t know true population parameters *ALWAYS some probability we are wrong

P(killed by lightning) 1/1,000,000 p = .000001

P(win powerball jackpot) 1/100,000,000 ~

Actual state of nature

H0 is true H0 is false

Decision

Reject H0

Correct

CorrectType I Error

Type II Error

Errors

Accept H0

Definitions & Symbols

Level of significance Probability of Type I error

1 - Level of confidence

Probability of Type II error

1 - Power ~

Steps in Hypothesis Test

1. State null & alternative hypotheses

2. Set criterion for rejecting H0

3. Collect sample; compute sample statistic & test statistic

4. Interpret resultsis outcome statistically significant? ~

Example: Nondirectional Test

Experimental question: Does reading to young children affect IQ scores?

= 100, = 15, n = 25 We will use z test

Same as computing z scores for ~X

Step 1: State Hypotheses

H0: = 100 Reading to young children will not

affect IQ scores. H1: 100

Reading to young children will affect IQ scores. ~

2. Set Criterion for Rejecting H0

Determine critical value of test statistic defines critical region(s)

Critical region also called rejection region

area of distribution beyond critical value in tails

If test statistic falls in critical regionReject H0 ~

2. Set Criterion for Rejecting H0

Level of Significance () Specifies critical region

area in tail(s) Defines low probability sample means

Most common: = .05 others: .01, .001

Critical value of z use z table for level ~

Critical Regions

f

+1 +20-1-2

+1.96-1.96

= .05

zCV = + 1.96

3. Collect data & compute statistics Compute sample statistic

Observed value of test statistic

Need to calculate ~

X

X

X

obs

Xz

3. Collect sample & compute statistics

3

1005.105 83.1

15100 ,

nX

25

15 3

3

5.5

n = 25 : 105.5assume X

X

obs

Xz

Critical Regions

f

+1 +20-1-2

+1.96-1.96

= .05

zCV = + 1.96

4. Interpret Results

Is zobs in the critical region? NO we fail to reject H0

These data suggest reading to young children does not affect IQ.

No “significant” difference does not mean they are equal

data inconclusive ~