Introduction to Hypothesis Testing for μ Research Problem: Infant Touch Intervention Designed to increase child growth/weight Weight at age 2: Known population:

Introduction to HypothesisTesting for μ

Research Problem: Infant Touch Intervention

Designed to increase child growth/weightWeight at age 2:

Known population: μ = 26 σ = 4

Sample data:n = 16 = 30

Did intervention increase weight?

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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 μ

Can’t take difference at face value

Differences between and μ expected simply on the basis of chance

sampling variability

How do we know if it’s just chance?

Sampling distributions!

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Research Problem:

Infant Touch Intervention

Known population: μ = 26 σ = 4

Assume intervention does NOT affect weight

Sample means ( ) should be close to population μ

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Compare Sample Data to know population:

z-test =

How much does deviate from μ?

What is the probability of this occurrence?

How do we determine this probability?

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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

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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 no effect

(b) Scientific Hypothesis: H1

There is an effect

Also called alternative hypothesis

Form of Ho and H1 for one-sample mean:

H0 : μ = 26

H1 : μ <> 26

Always about a population parameter, not a statistic

H0 : μ = population value

H1 : μ <> population value

non-directional (two-tailed) hypothesis

mutually exclusive :cannot both be true

Example:

Infant Touch Intervention

Known population:μ = 26 σ = 4

Did intervention affect child weight?

Statistical Hypotheses:

H0: μ = 26

H1 : μ <> 26

Hypothesis Testing Steps:

(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 distributionDivide 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?

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Level of significance = alpha level = α

Probability chosen as criteria for “unlikely”Common convention: α = .05 (5%)

Critical value = boundary between likely/unlikely outcomesCritical 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

Hypothesis Testing Steps:

(4) Collect data and Calculate “observed” test statistic

z-test for one sample mean:

A closer look at z: z = sample mean – hypothesized population μ

standard error

z = observed difference difference due to chance

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z

nx

Hypothesis Testing Steps:

(5) Make a decisionTwo 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” findingIf NO,

Fail to Reject H0 = “non-significant” finding

Hypothesis Testing Steps:

(6) Interpret results

Return to research question

statistical significance = not likely to be due to chance

Never “prove” or H0 or H1

Example (1) Does touch increase weight?Population: μ = 26 σ = 4

(2) Statistical Hypotheses: H0 : μ = H1 : μ <>

(3) Decision Rule:α = .05

Critical value:

(4) Collect sample data: n = 16 = 30

Compute z-statistic:

(5) Make a decision:

(6) Interpret results:Intervention appears to increase weight. Difference not likely to be due to chance.

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z

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nx

More about alpha (α) levels:

most common : α = .05

more stringent : α = .01

α = .001

Critical values for two-tailed z-test:

α = .05 α =.01 α =.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 α in half:

probability in each tail = α / 2

p=.025 p=.025

1.96 +1.96

= .05

B. One-tailed (directional):

H0: 26

H1 : > 26

Upper tail critical:

H0 : 26

H1 : < 26

Lower tail critical:

+1.65

p=.05

z

1.65

p=.05

z

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 (α)?α = alpha level.05 .01 .001

(b)Is H0 one-tailed or two-tailed?If one-tailed: find CV for α

CV will be either + or -If two-tailed: find CV for α /2

CV will be both +/ -

Most Common choices:• α = .05 • two-tailed test

Commonly used Critical Values

for the z-statisticHypothesis α = .05 α =.01______________________________________________Two-tailed 1.96 2.58 H0: = x H1: xOne-tailed upper + 1.65 + 2.33 H0: x H1: > xOne-tailed lower 1.65 2.33 H0: x H1: < x______________________________________________Where x = any hypothesized value of under H0

Note: critical values are larger when:a more stringent (.01 vs. .05)test is two-tailed vs. one-tailed

III. Outcomes of Hypothesis Testing

Four possible outcomes:

True status of H0 No Effect Effect

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 errorα = level of significance

p (Type I error)1- α = level of confidence

p(correct decision), when H0 true

if α = .05, confidence = .95if α = .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 (α)Choose α = .01 or .001 if error would be very

seriousOtherwise, α = .05 is small but reasonable risk

B. Assume Ho is false

Only one mistake is relevant Type II error

= probability of Type II error1- = ”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” isPower = 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 a level

e.g., choose .05 vs. .01

One-tailed vs. two-tailed tests

Four Possible Outcomes of an Hypothesis Test

True status of H0 H0 true H0 false

Reject H0

Decision Retain H0

α = level of significanceprobability of Type I Error

risk of rejecting a true H0

1- α = level of confidencep (making correct decision), if H0 true

= probability of Type II Error risk of retaining a false H0

1- = powerp(making correct decision), if H0 falseability to detect true effect

1-

Type I Error Power

1- Confidence Type II Error

IV. Additional Comments

A.Statistical significance vs. practical significance

“Statistically Significant” = H0 rejectedB.Assumptions of the z-test (see book

for review):

DSM is normal Known (and unaffected by treatment)Random samplingIndependent observationsRare to actually know !Preview use t statistic when unknown

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hypz

V. Reporting Results of an Hypothesis Test

If you reject H0:

“There was a statistically significant difference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 30 lbs), z = 4.0, p < .05, two-tailed.”

If you fail to reject H0 :

“There was no significant difference in weight between children in the intervention sample (M = 30 lbs) and the general population (M = 30 lbs), z = 1.0, p > .05, two-tailed.”

A closer look…

z = 4.0, p < .05

test statistic

observed value

level of significance

VI. Effect SizeStatistical significance vs. practical importanceHow large is the effect, in practical terms?Effect size = descriptive statistics that indicate the magnitude of an effect

Cohen’s dDifference between means in standard deviation units

Guidelines for interpreting Cohen’s d

Effect Size d

Small .20

Medium .20 < d .80

Large d > .80