Transcript Hypothesis Testing I

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
systematicvariation
test statistic
unsystematic variation
difference between groups effect
test statistic 

difference due to chance
error
zobs 
X 
X
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

statistical significance


by data
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 (a)
 incorrectly rejecting Ho
 e.g., may think a new antidepressant is
effective, when it is NOT ~

Errors
Type II error (b)
 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 ~
Errors
Actual state of nature
H0 is true
Reject H0
H0 is false
Type I
Error
Correct
Correct
Type II
Error
Decision
Accept H0
Definitions & Symbols

a
Level of significance
 Probability of Type I error
1-a
 Level of confidence


b
Probability of Type II error
1-b
 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 results
is 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

area of distribution beyond critical value
in

called rejection region
tails
If test statistic falls in critical region
Reject H0 ~
2. Set Criterion for Rejecting H0

Level of Significance (a)
 Specifies critical region
area

in tail(s)
Defines low probability sample means
 Most common: a = .05
others:

.01, .001
Critical value of z
 use z table for a level
~
Critical Regions
a = .05
zCV = + 1.96
f
-2
-1.96
-1
0
+1
+2
+1.96
3. Collect data & compute statistics

Compute sample statistic
X

Observed value of test statistic
zobs 

X 
X
Need to calculate  X
~
3. Collect sample & compute statistics
  100,   15
assume : X  105.5
X 
zobs 

n
X 
X
n = 25
15
3

25
105 .5  100
5.5
 1.83


3
3
Critical Regions
a = .05
zCV = + 1.96
f
-2
-1.96
-1
0
+1
+2
+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 ~