test of (µ 1 – µ 2 ), 1 = 2, populations normal test statistic and df = n 1 + n 2 – 2 2–...

Test of (µ1 – µ2), 1 = 2, Populations Normal

• Test Statistic

and df = n1 + n2 – 2

2–21

22

)1–2

( 21

)1–1

( 2 where

21

112

0]

2–

1[– ]

2–

1[

nn

snsnps

nnps

xxt

© 2008 Thomson South-Western

• Test Statistic

Test of (µ1 – µ2), Unequal Variances, Independent Samples


1)(

1)(

)()( where

)()(

2

2

2

2

2

1

2

1

2

1

2

2

2

21

2

1

2

2

2

1

2

1

02121

nns

nns

nsnsdf

ns

ns

xxt

Test of Independent Samples(µ1 – µ2), 1 2, n1 and n2 30

• Test Statistic

– with s12 and s2

2 as estimates for 12 and

22

z [x

1– x

2]–[

1–

2]0

s12

n1

s2

2n2


Test of Dependent Samples(µ1 – µ2) = µd

• Test Statistic

– where d = (x1 – x2)

= d/n, the average differencen = the number of pairs of

observationssd = the standard deviation of d

df = n – 1

nd

sdt

d


Test of (1 – 2), where n1p15, n1(1–p1)5, n2p25, and n2 (1–p2 )

• Test Statistic

– where p1 = observed proportion, sample 1

p2 = observed proportion, sample 2

n1 = sample size, sample 1

n2 = sample size , sample 2p

n1

p1

n2

p2

n1

n2

zp p

p p n n

1 2

1 11

12

( )


Test of 12 = 2

2

• If 12 = 2

2 , then 12/2

2 = 1. So the hypotheses can be worded either way.

• Test Statistic: whichever is

larger • The critical value of the F will be F(/2, 1, 2)

– where = the specified level of significance1 = (n – 1), where n is the size of the

sample with the larger variance2 = (n – 1), where n is the size of the sample

with the smaller variance

21

22 or

22

21

s

s

s

sF


Confidence Interval for (µ1 – µ2)

• The (1 – )% confidence interval for the difference in two means:– Equal-variances t-interval

– Unequal-variances t-interval

2

1

1

122

)2

–1

(nnpstxx

2

22

1

21

2 )

2–

1(

n

s

n

stxx


Confidence Interval for (µ1 – µ2)

• The (1 – )% confidence interval for the difference in two means:– Known-variances z-interval


2

2

2

1

2

1221 )(

nnzxx

Confidence Interval for (1

– 2) • The (1 – )% confidence interval for the difference in two proportions:

– when sample sizes are sufficiently large.

(p1

– p2

) z2

p1(1– p

1)

n1

p2

(1– p2

)

n2


One-Way ANOVA, cont.• Format for data: Data appear in separate

columns or rows, organized by treatment groups. Sample size of each group may differ.

• Calculations:– SST = SSTR + SSE (definitions

follow)

– Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean, , across all data... total variation in the data (not variance).2)–( SST xijx

x


One-Way ANOVA, cont.• Calculations, cont.:

– Sum of squares treatment (SSTR) = sum of squared differences between each group mean and the grand mean, balanced by sample size... between-groups variation (not variance).

– Sum of squares error (SSE) = sum of squared differences between the individual data values and the mean for the group to which each belongs... within-group variation (not variance).

2)–( SSTR xjxj

n

SSE (xij

– x j)2


One-Way ANOVA, cont.• Calculations, cont.:

– Mean square treatment (MSTR) = SSTR/(t – 1) where t is the number of treatment groups... between-groups variance.

– Mean square error (MSE) = SSE/(N – t) where N is the number of elements sampled and t is the number of treatment groups... within-groups variance.

– F-Ratio = MSTR/MSE, where numerator degrees of freedom are t – 1 and denominator degrees of freedom are N – t.


Goodness-of-Fit Tests• Test Statistic:

where Oj = Actual number observed in each class

Ej = Expected number, j • n

jE

jEjO 2)–( 2


Chi-Square Tests of Independence• Hypotheses:– H0: The two variables are independent.

– H1: The two variables are not independent.

• Rejection Region:– Degrees of freedom = (r – 1) (k – 1)

• Test Statistic:

ijEijEijO 2)–(

2


Chi-Square Tests of Multiple ’s• Rejection Region: Degrees of freedom: df = (k – 1)

• Test Statistic:

2 (O

ij–E

ij)2

Eij


Determining the Least Squares Regression Line• Least Squares Regression Line:

– Slope

– y-intercept

ˆ y b0

b1x1

b1

( x

iyi) – nx y

( xi2) – nx 2

b0

y – b1x


To Form Interval Estimates

• The Standard Error of the Estimate, sy,x

– The standard deviation of the distribution of the»data points above and below the regression

line,»distances between actual and predicted

values of y,» residuals, of

– The square root of MSE given by ANOVA2–

2)ˆ–( , n

yiyxys


Equations for the Interval Estimates• Confidence Interval for the Mean of y

• Prediction Interval for the Individual y

nix

ix

xvaluexnxysty

2)(– )2(

2)– ( 1),(2

ˆ

ˆ y t2(sy,x) 1 1n (x value – x )2

( xi2) –

( xi)2

n


Coefficient of Correlation, r and Coefficient of Determination, r2

i i i i

2 2 2 2i i i i

n( x y ) ( x )( y )r 0.679

n( x ) ( x ) * n( y ) ( y )

Three Tests for Linearity• 1. Testing the Coefficient of Correlation

H0: = 0 There is no linear relationship between x and y.H1: 0 There is a linear relationship between x and y.

Test Statistic:

• 2. Testing the Slope of the Regression LineH0: = 0 There is no linear relationship between x and y.H1: 0 There is a linear relationship between x and y.

Test Statistic:

t r1 – r2n – 2

tb

sy xx n x

1

2 2,

( )© 2008 Thomson South-Western

Three Tests for Linearity• 3. The Global F-test

H0: There is no linear relationship between x and y.

H1: There is a linear relationship between x and y.

Test Statistic:

Note: At the level of simple linear regression, the global F-test is equivalent to the t-test on 1. When we conduct regression analysis of multiple variables, the global F-test will take on a unique function.

F MSRMSE

SSR

1SSE

(n – 2)


A General Test of 1• Testing the Slope of the Population

Regression Line Is Equal to a Specific Value.H0: =

The slope of the population regression line is .

H1:

The slope of the population regression line is not .

Test Statistic:2)(– 2

,10

– 1

xnx

xysb

t


test of (µ 1 – µ 2 ), 1 = 2, populations normal test statistic and df = n 1 + n 2 – 2 2–...

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