Introduction to Hypothesis
Testing for
Research Problem:
Infant Touch Intervention
Designed to increase child growth/weight
Weight at age 2:
Known
population: = 26 = 4
Sample data:n = 16
= 30
Did intervention increase weight?
Hypothesis Testing:
- Using sample data to evaluate an hypothesis about a population parameter.
- Usually in the context of a research study evaluate effect of a “treatment”
- Compare to known
- Can’t take difference at face value
- Differences between and expected simply on the basis of chance
sampling variability
- How do we know if it’s just chance?
Sampling distributions!
Research Problem:
Infant Touch Intervention
Known population: = 26 = 4
- Assume intervention does NOT affect weight
- Sample means () should be close to population
Compare Sample Data to know population:
z-test =
How much does deviate from ?
What is the probability of this occurrence?
How do we determine this probability?
Distribution of Sample Means (DSM)!
in the tails are low probability
How do we judge “low” probability of occurrence?
Widely accepted convention.....
5 in a 100 p .05
Logic of Hypothesis Testing
- Rules for deciding how to decide!
- Easier to prove something is false
Assume opposite of what you believe…
try to discredit this assumption….
Two competing hypotheses:
(1)Null Hypothesis ( H0 )
The one you assume is true
The one you hope to discredit
(2)Alternative Hypothesis ( H1 )
The one you think is true
Inferential statistics:
Procedures revolve around H0
Rules for deciding when to reject or retain H0
Test statistics or significance tests:
- Many types z-test t-test F-test
- Depends on type of data and research design
- Based on sampling distributions, assumes H0 is true
- If observed statistic is improbable given H0,
then H0 is rejected
Hypothesis Testing Steps:
(1)State the Research Problem
Derived from theory
example:
Does touch increase child growth/weight?
(2)State statistical hypotheses
Two contradictory hypotheses:
(a)Null Hypothesis: H0
There is noeffect
(b)Scientific Hypothesis: H1
There is an effect
Also called alternative hypothesis
Form of Ho and H1 for one-sample mean:
H0: = 26
H1: 26
- Always about a population parameter, not a statistic
H0: = population value
H1: population value
- non-directional (two-tailed) hypothesis
- mutually exclusive cannot both be true
Example:
Infant Touch Intervention
Known population: = 26 = 4
Did intervention affect child weight?
Statistical Hypotheses:
H0: = 26
H1: 26
(3)Create decision rule
- Decision rule revolves around H0, not H1
- When will you reject Ho?
…when values of are unlikely given H0
- Look in tails of sampling distribution
- Divide distribution into two parts:
Values that are likely if H0 is true
Values close to H0
Values that are very unlikely if H0 is true
Values far from H0
Values in the tails
How do we decide what is likely and unlikely?
Level of significance = alpha level =
Probability chosen as criteria for “unlikely”
Common convention: = .05 (5%)
Critical value = boundary between likely/unlikely outcomes
Critical region = area beyond the critical value
Decision rule:
Reject H0 when observed test-statistic (z) equals or exceeds the Critical Value (when z falls within the Critical Region)
Otherwise, Retain H0
(4)Collect data and Calculate “observed” test statistic
z-test for one sample mean:
where, =
A closer look at z:
z = sample mean – hypothesized population
standard error
z = observed difference
difference due to chance
(5)Make a decision
Two possible decisions:
Reject H0
Retain (Fail to Reject) H0
Does observed z equal or exceed CV?
(Does it fall in the critical region?)
If YES,
Reject H0 = “statistically significant” finding
If NO,
Fail to Reject H0 = “non-significant” finding
(6)Interpret results
Return to research question
statistical significance = not likely to be due to chance
Never “prove” H0 or H1
Example
(1)Does touch increase weight?
Population: = 26 = 4
(2)Statistical Hypotheses:
H0:
H1:
(3)Decision Rule:
= .05
Critical value:
(4)Collect sample data: n = 16 = 30
Compute z-statistic:
z = =z =
(5)Make a decision:
(6)Interpret results:
Reporting Results of an Hypothesis Test
If you reject H0:
“There was a statistically significantdifference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 26 lbs),z = 4.0, p < .05, two-tailed.”
If you fail to reject H0:
“There was nosignificantdifference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 26 lbs),z = 1.0, p > .05, two-tailed.”
A closer look…
z = 4.0, p < .05
Alternative statistical findings:
Significant z = 4.0, p < .01
Significant z = 4.0, p < .001
Non-significant z = 4.0, p > .05
Non-significant z = 4.0,p > .01
More about alpha () levels:
most common = .05
more stringent = .01
= .001
Critical values for two-tailed z-test:
= .05 / =.01 / =.001 1.96 / 2.58 / 3.30
More About Hypothesis Testing
I.Two-tailed vs. One-tailed hypotheses
A.Two-tailed (non-directional):
H0: = 26
H1: 26
Region of rejection in both tails:
Divide in half:
probability in each tail =
B.One-tailed (directional):
H0: 26
H1: > 26
Upper tail critical:
H0: 26
H1: < 26
Lower tail critical:
Examples:
Research hypotheses regarding IQ, where hyp= 100
(1)Living next to a power station will lower IQ?
H0:
H1:
(2)Living next to a power station will increase IQ?
H0:
H1:
(3)Living next to a power station will affect IQ?
H0:
H1:
When in doubt, choose two-tailed!
II.Selecting a critical value
Will be based on two pieces of information:
(a)Desired level of significance ()?
=alpha level
.05 .01 .001
(b)Is H0 one-tailed or two-tailed?
If one-tailed: find CV for
CV will be either + or -
If two-tailed: find CV for /2
CV will be both +/ -
Most Common choices:
- = .05
- two-tailed test
Commonly used Critical Values
for the z-statistic
______
Hypothesis = .05=.01
______
Two-tailed 1.96 2.58
x
x
One-tailed upper+ 1.65+ 2.33
x
x
One-tailed lower 1.65 2.33
x
x
______
Where x = any hypothesized value of under H0
Note: critical values are larger when:
- more stringent (.01 vs. .05)
- test is two-tailed vs. one-tailed
III.Outcomes of Hypothesis Testing
Four possible outcomes:
True status of H0
H0 true H0 false
Reject H0
Decision
Retain H0
Type I Error:Rejecting H0 when it’s actually true
Type II Error:Retaining H0 when it’s actually false
We never know the “truth”
Try to minimize probability of making a mistake
A.Assume Ho is true
Only one mistake is relevant Type I error
level of significance
p (Type I error)
1-level of confidence
p(correct decision), when H0 true
if = .05, confidence = .95
if = .01, confidence = .99
So, mistakes will be rare when H0 is true!
How do we minimize Type I error?
WE control error by choosing level of significance ()
Choose = .01 or .001 if error would be very serious
Otherwise, = .05 is small but reasonable risk
B.Assume Ho is false
Only one mistake is relevant Type II error
probability of Type II error
1-”Power”
p(correct decision), when H0 false
How big is the “treatment effect”?
When “effect size” is big:
Effect is easy to detect
is small (power is high)
When “effect size” is small:
Effect is easy to “miss”
is large (power is low)
How do you determine and power (1-)
- No single value for any hypothesis test
- Requires us to guess how big the “effect” is
Power =probability of making a correct decision
when H0 is FALSE
C.How do we increase POWER?
Power will be greater (and Type II error smaller):
- Larger sample size (n)
Single best way to increase power!
- Larger treatment effect
- Less stringent level
e.g., choose .05 vs. .01 - One-tailed vs. two-tailed tests
Caution – don’t choose this after the fact
Four Possible Outcomes of an Hypothesis Test
True status of H0
H0 true H0 false
Reject H0
Decision
Retain H0
= level of significance
probability of Type I Error
risk of rejecting a true H0
1- =level of confidence
p (making correct decision), if H0 true
= probability of Type II Error
risk of retaining a false H0
1- = power
p(making correct decision), if H0 false
ability to detect true effect
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