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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
( )
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
Confidence Interval for (µ1 – µ2)
• The (1 – )% confidence interval for the difference in two means:– Known-variances z-interval
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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.
© 2008 Thomson South-Western
Goodness-of-Fit Tests• Test Statistic:
where Oj = Actual number observed in each class
Ej = Expected number, j • n
jE
jEjO 2)–( 2
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
Chi-Square Tests of Multiple ’s• Rejection Region: Degrees of freedom: df = (k – 1)
• Test Statistic:
2 (O
ij–E
ij)2
Eij
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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)
© 2008 Thomson South-Western
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
© 2008 Thomson South-Western
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