x±z*(σn)

n=(z*σME)2

Reduce ME

1. Lower confidence

2. reduce σ

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3. increase n

HT for pop mean

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z=x-μσ√n

Errors

I=P(reject H0/H0 true)=α

II=P(accept H0/H0 wrong)

Power=1-β

Increase pow

1. increase α

2. consider alternative µ that is farther than hypothesized µ 3.increase n

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4.decrease σ

One sample t CI

x±t*s√n

One sample t-test

T=x-µ0s√n

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Df=n-1

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P is 2 closest

Two pop means (before and after)

xD±t*sD√nD

xD=x1-x2

sD=x2-(x)2nn-1

Hypothesis test

H0:µa=µb

H1: µa≠µb

T=xd-D0Sd√n

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Df=n-1

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P is 2 closest

Two pop means (2 samples one condition)

σ known

CI:

xd±z*σ12n1+σ22n2

Hypothesis test:

H0:µa=µb

H1: µa≠µb

Z=x1-x2-Dσ12n1+σ22n2

s-known

CI:

xd±t*s12n1+s22n2

Hypothesis test:

t=x1-x2-Ds12n1+s22n2

df= lower of (ni-1)

σ unknown (σ1=σ2)

CI:

xd±t*sp21n1+1n2

sp2=sampled pool variance

df=n1+n2-2

Hypothesis test:

T=x1-x2sp21n1+1n2

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sp2=n1-1s12+n2-1s22n1+n2-2

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P is 2 closest

Compare 2 pop variances

Assumptions:

1.2 pops are normally dist

2. samples are random and indep

H0:σ12=σ12

H1: σ12≠σ12

F=s22s12

(or f=s12s22 when H1:σ12>σ22)

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f: numerator degrees of freedom and denominator degrees of freedom (n-1) Single proportion

p=sample prop=X/n

(X= number of success)

p0=pop prop= S/n

CI:

p±z*p(1-p)n

Hypothesis test

H0:p=p0

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Z=p-p0p(1-p)n

Two pop proportions

CI:

p1-p2±z*p1(1-p1)n1+p2(1-p2)n2

Hypothesis test:

H0:p1=p2

Z=p1-p2p(1-p)(1n1+1n2)

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p=x1+x2n1+n2

Chi-square (indep test)

Hypothesis test:

H0:no association between row and column variables

X2=o-e2e

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df=(#rows-1)(#col-1)

Least squares regression line

Deviation assumptions:

1. independent

2. normally distributed

3. mean=0

4.SD of s

Inference for regression slope

using sample slope to make inference about pop slope

Assumptions:

1. SRS from pop

2. there is a linear relationship in pop

3.SD of the responses about the population line is the same for all values of the explanatory variables 4. response variables varies normally about the pop regression line. CI for slope(β1):

b1±t*sb1

sb1=sSSxx

SSxx=sum of squares

Prediction interval for individual (wider than confidence for mean) ŷ±t*se1+1n+x-x2SSxx

or

ŷ±t*SEŷ

df=n-2

CI for a mean response:

ŷ±t*se1n+x-x2SSxx

or

ŷ±t*SEµ

Hypothesis test:

H0: b1=0 (model has no predictive value)

-if asking for direct relationship use UTT

t=b1sb1

df=n-2

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P value is 2 closest values

Correlation

Hypothesis test:

H0:p=0

T=rn-21-r2

r=correlation

df=n-2

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P value is 2 closest values

Multiple regression

cond’s for error variable ε

1. the prob of distribution of ε is normal

2. the mean of ε is 0

3. th SD of ε is σε, which is constant for each value of x 4. errors are independent

residuals:

ei=yi-ŷi (how far is your value true value)

hypothesis test for individual variable:

H0:βi=0

t=b1sb1

df=n-k-1

P value is 2 closest values

Hypothesis test for entire model:

H0:β1+β2+…+βk=0

H1: at east one βi≠0

F=MSRMSE

df: numerator=k

denominator=n-k-1

Sε=SSEn-k-1

Sε=standard error of estimate

- if Sε is high then there is a lot of variability

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R2=SSR/SST

ANOVA

Hypothesis:

H0:µ1=µ2=…=µk

H1: at least one µi is different

F=MSTMSE

df:...