Evaluating Hypotheses

Chapter 9

Descriptive vs. Inferential Statistics

Descriptive quantitative descriptions of

characteristics

Inferential Statistics

Making conclusions (inferences) about parameters e.g., X confidence intervals: infer lies

within interval also quantitative ~

Hypothesis Testing

Most widely used inferential statistics Hypothesis

testable assumption or inference about a parameter or distribution

should conclusion (inference) be accepted

final result a decision: YES or NO qualitative not quantitative ~

Hypothesis Testing

Example: IQ scores = 100, = 15 Take random sample of students

n = 10 Hypothesis:

sample is consistent with population with above parameters

sample is the same as population ~

Evaluating Hypotheses

Test statement about population using a statistic X for a sample:

add values & divide by n impossible or difficult for population need rules based on properties of

samples ~

Evaluating Hypotheses

Proving / Disproving Hypotheses

Logic of science built on disproving easier than proving but ultimately want to prove

State 2 mutually exclusive hypotheses if one is true, other cannot be true ~

Steps in Hypothesis Evaluation

1. State null & alternative hypothesesH0 and H1

2. Set criterion for rejecting H0

level of significance:

3. collect sample; compute sample statistic & test statistic

4. Interpret resultsis outcome statistically significant? ~

Hypothesis Evaluation

1. Null Hypothesis: H0

there is no difference between groups

2. Alternative Hypothesis: H1 there is a difference between

groups ~

Hypothesis Evaluation

Example: IQ and electric fields question: Does living near power lines affect IQ

of children? H0 : there is no difference

Living near power lines does not alter IQ. = 100

H1 : Living near power lines does alter IQ. 100 ~

Hypothesis Evaluation

Outcome of study reject or “accept” null hypothesis

Reject Ho

accept as H1 true “Accepting” null hypothesis

difficult or impossible to “prove” Ho

actually: fail to reject Ho

do not have enough evidence to reject ~

Evaluating Ho and H1

Hypotheses about population parameters

Test statistic especially designed to test Ho

Procedure depends on… particular test statistic used directionality of hypotheses level of significance ~

Directionality & Hypotheses

Directionality effects critical values used Nondirectional

two-tailed test Ho : = 100; H1 : 100 change could be either direction Do not know what effect will be

may increase or decrease values ~

Directionality & Hypotheses

Directional one tailed Have prior evidence that suggests

direction of effectpredict that effect will be larger

or smaller, but only 1

Ho: < 100

H1: > 100 ~

Errors

“Accept” or reject Ho

only probability we made correct decision also probability made wrong decision

Type I error rejecting Ho when it is really true e.g., may think a new antidepressant is

effective, when it is NOT ~

Errors

Type II error “accepting” Ho when it is really false e.g., may think a new antidepressant is

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

because we do not know true population parameters ~

Actual state of nature

H0 is true H0 is false

Decision

Accept H0

Reject H0

Correct

CorrectType I Error

Type II Error

Errors

Level of Significance ()

Probability of making Type I error complement of level of confidence .95 + .05 = 1

= .05 conduct experiment 100 times 5 times will make Type I error

rejected H0 when it should be accepted

Want probability of Type I error small ~

Statistical Significance

If reject H0

Outcome is “statistically significant” difference between groups is ...

greater than expected by chance alone due to sampling, etc.

Does NOT say it is meaningful ~

Statistical Power

Power probability of correctly rejecting H0

= probability of type II error complement of power ~

Practical Significance Degree to which result is important

result can be statistically significant but not important in real world no practical implications no universal method for reporting

Effect size measure of magnitude of result difference between means of 2 groups e.g., IQ: 1 point small effect, 15 large ~

Procedure for Evaluating Hypotheses

Experiment Draw random sample compute statistic determine if reasonably comes from

populationIf no reject H0

Use test statistic to make decision 3 important distributions

variable, sample statistic, test statistic~

Test Statistic distribution of test statistic

has known probabilities General form

test statistic = sample statistic - population parameter

standard error of sample statistic

difference actually obtainedX -

divided by difference by chance alone ~

Steps in Hypothesis Evaluation

1. State null & alternative hypothesesH0 and H1

2. Set criterion for rejecting H0

level of significance:

3. collect sample; compute sample statistic & test statistic

4. Interpret resultsis outcome statistically significant? ~