Transcript Evaluating Hypotheses

Evaluating Hypotheses
Chapter 9
Descriptive vs. Inferential Statistics

Descriptive
 quantitative descriptions of
characteristics
Inferential Statistics

Making conclusions (inferences)
about parameters
 e.g., m  X
 confidence intervals: infer m 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
 m = 100, s = 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 hypotheses
H0 and H1
2. Set criterion for rejecting H0
level of significance: a
3. collect sample; compute sample
statistic & test statistic
4. Interpret results
is 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.
 m = 100
 H1 : Living near power lines does alter IQ.
 m  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 : m = 100; H1 : m  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 effect
predict that effect will be larger
or smaller, but only 1


Ho: m < 100
H1: m > 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 ~

Errors
Actual state of nature
H0 is true
Accept H0
Correct
Reject H0
Type I
Error
H0 is false
Type II
Error
Decision
Correct
Level of Significance (a)
Probability of making Type I error
 complement of level of confidence
 .95 + .05 = 1
 a = .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
 b = 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
population
If 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 obtained
X-m

divided by difference by chance alone ~
Steps in Hypothesis Evaluation
1. State null & alternative hypotheses
H0 and H1
2. Set criterion for rejecting H0
level of significance: a
3. collect sample; compute sample
statistic & test statistic
4. Interpret results
is outcome statistically significant? ~