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Yes – No questions

1. Is the IQ-level of university students above average?

2. Is there a difference between verbal IQ-level of males and females?

3. Can we learn in complete silence better than with a moderately loud music?

4. Is there a relationship between blood pressure and the Tolerance scale of CPI?

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X=WAIS-IQ, Population = University students H0: E(X) = 100

H1: E(X) < 100

H2: E(X) > 100

H0: Med(X) = 100

H1: Med(X) < 100

H2: Med(X) > 100

H0: E(X) = 100

HA: E(X) 100

H0: Med(X) = 100

HA: Med(X) 100

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Verbal intelligence: WAIS/VIQ, E(VIQ/female) = f E(VIQ/male) = m

H0: f = m

H1: f < m

H2: f > m

H0: f = m

HA: f m

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The above hypotheses always refer to the population characteristics of the variables studied (expected value, median, etc.)

Only one of them can be true at the same time.

H0, the null hypothesis, can only be true in one single case. The alternative hipotheses can be true in infinite ways.

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X-sample

H0 H1 H2

Which of them is true?

Statistical test

What is a statistical test?

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A procedure for testing theH0: E(X)=100 hypothesis

1. Interval estimation for E(X): C0.95 = (c1; c2).X

c2c1

2. E(X) is likely between c1 and c2.

3. If 100 is also between c1 and c2, keep H0.

4. If c2 < 100, accept H1: E(X) < 100.

5. If c1 > 100, accept H2: E(X) > 100.

100? 100?100?

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X-sample

H1: E(X) < 0

H0 H2: E(X) > 0

H0: E(X) = 0

n

Xz =

0

z -1.96 z 1.96

z test Assumptions: X is normally distributed, is known

0.95

1.96-1.96

N(0,1)

0.0250.025

|z| < 1.96

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Testing the H0: E(X) = hypothesis if is unknown

If H0: E(X) = is true, then the test statistic

s n

Xt =

follows a t distribution with df = n -1, provided that X is normal, or n is large. Since |t| < t0,05

with a probability of 95%, |t| t0,05 is very unlikely. If despite this it occurs, it is an evidence against H0, in which case we reject .

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X-sample

H1: E(X) <

H0 H2: E(X) >

H0: E(X) =

t -t0.05

t t0.05

|t| < t0.05

One-sample t-test

s n

Xt =

Assumption: X is normal

95

t

25 25

t0.05-t 0.05

1

Basic terms of two-tailed tests illustrated withthe t-test (level of significance = )

195

t

225 225

t0.05-t0.05

Region of acceptance

Criticalregion

Criticalregion Critical values

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Basic terms of one-tailed tests illustrated withthe t-test (level of significance

195

t

5

t0.10

Region of acceptanceCriticalregionCritical value

H0: E(X) =

H2: E(X) >

Assumption:H1: E(X) <

is equiv. to H0

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Errors in a statistical test If we reject H0:

– Error: false rejection– Name of error: Type I error– Probability: level of significance ()– Effect: test validity

If we keep H0:– Error: false acceptance– Name of error: Type II error– Probability: generally not known ()– Effect: test efficiency

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Comparing two variables or populations

1. Is there any difference between verbal and performance IQ-levels at schizophrenics?

2. Is body temperature larger in the morning than in the evening?

3. Is the tolerance level of neurotics smaller than that of psychopaths?

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Comparing two means

Examples: H0: E(VIQ/Sch) = E(PIQ/Sch)

H0: E(Morning temp.) = E(Evening temp.)

H0: E(CPI-Tol/Neurot) = E(CPI-Tol/Ppath)

Often (if X and Y are quantitative):H0: 1 = 2

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Case of one population, two variables

Example: comparing VIQ-level and PIQ-level. Solution: Z = VIQ-PIQ, or (only at ratio scaled

variables) Z = Y/X New null hypothesis:

H0: E(Z) = 0 or H0: E(Z) = 1 Statistical test: one sample t-test. Steps: draw a random sample, compute z-scores,

perform one-sample t-test on the Z-sample.

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Example: Comparing verbal IQ of males and females.

Null hypothesis: H0: 1 = 2

Sampling: Draw two independent samples from the two populations.

Comput.: Compute sample means and variances:Sample size Mean Variance

Sample 1: n1 x1 var1= (s1)2

Sample 2: n2 x2 var2= (s2)2

Case of two populations, one variable

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The two-sample t-testIf the H0: 1 = 2 null hypothesis is true,

and X is normal and 1 = 2, then the

t X X

e eVar

n

Var

n

1 2

1 2

statistic follows a t distribution with df = f1 + f2, where f1 = n1-1, f2 = n2-1, and

eVarf Var f Var

f f

1 1 2 2

1 2

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X-sample

H1: 1 < 2 H0 H2: 1 >

2

Assumptions: normality,independent samples,1 = 2

t -t0.05

t t0.05

|t| < t0.05

Two-sample t-test

tVar

n

Var

n

X X

e e

1 2

1 2

H0: 1 =

2

95

t

25 25

t0.05-t 0.05

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The Welch-testIf the H0: 1 = 2 null hypothesis is

true and X is normal, then the

test statistic follows approximately a t distribution with the following df (a = Var1/n1, b = Var2/n2):

fa b

a

f

b

f

2

2

1

2

2

( )

2

2

1

1

y'

n

Var

n

Var

xt

2

X-sample

H1: 1 <

2

H0 H2: 1 >

2

H0: 1 =

2

t’ -t0.05

|t’| < t0.05

The Welch-test Assumptions: normality, two independent samples

t’ t0.05

95

t

25 25

t0.05-t 0.05

2

2

1

1

y'

n

Var

n

Var

xt

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X-sample

H0: 1 = 2 HA: 1 2

H0: 1 =

2

F < F0.025

F F0.025

Fisher-s F test

F Var

Var max

min

Assumptions: normality, two independent samples

25

F0,025

975

F

1

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Robust statistical tests The Welch-test is a robust version of the

two-sample t-test, because it tests the same null hypothesis with fewer assumptions.

Robust versions of the F test (less sensitive to the violation of the normality assumption):Levene-testO’Brien-test