Download Chapter 10, Hypothesis Testing, Two Measures

Test of (µ1 – µ2), s1 = s2,
Populations Normal
• Test Statistic
[x – x ] – [m – m ]
t = 1 2  1 2 0


 1 + 1 
2
sp n n 


 1
2
(n –1)s 2 + (n –1)s 2
1
2
2
where s p2 = 1
n +n – 2
1 2
and df = n1 + n2 – 2
© 2008 Thomson South-Western
Test of (µ1 – µ2), Unequal
Variances, Independent Samples
• Test Statistic
( x1  x2 )  ( m1  m 2 )0
t=
s12 s22
+
n1 n2

(s
where df =
n1 ) + ( s n2 )
( s n1 ) 2 ( s22 n2 ) 2
+
n1  1
n2  1
2
1
2
1
2
2
2
© 2008 Thomson South-Western
Test of Independent Samples
(µ1 – µ2), s1 s2, n1 and n2  30
• Test Statistic
[x – x ]–[m – m ]
1 20
z = 1 2
s2 s 2
1 + 2
n n
1
2
– with s12 and s22 as estimates for s12 and s22
© 2008 Thomson South-Western
Test of Dependent Samples
(µ1 – µ2) = µd
• Test Statistic
t=s d
d
n
– where d = (x1 – x2)
d = Sd/n, the average difference
n = the number of pairs of observations
sd = the standard deviation of d
df = n – 1
© 2008 Thomson South-Western
Test of (p1 – p2), where n1p15,
n1(1–p1)5, n2p25, and n2 (1–p2 )
• Test Statistic
p p
1 2
z=


 1
p (1 p)  n + n1 

2 
 1
– where p1 = observed proportion, sample 1
p2 = observed proportion, sample 2
n1 = sample size, sample 1
n2 = sample size , sample 2
n p + n p
2 2
p = 1 1
n + n
1
2
© 2008 Thomson South-Western
Test of s12 = s22
• If s12 = s22 , then s12/s22 = 1. So the
hypotheses can be worded either way.
s2
s 2
• Test Statistic: F = 1 or 2 whichever is larger
s 2
s2
2
1
• The critical value of the F will be F(a/2, n1, n2)
– where a = the specified level of significance
n1 = (n – 1), where n is the size of the sample
with the larger variance
n2 = (n – 1), where n is the size of the sample
with the smaller variance
© 2008 Thomson South-Western
Confidence Interval for (µ1 – µ2)
• The (1 – a)% confidence interval for the
difference in two means:
– Equal-variances t-interval









1 1
(x – x )  t a  s 2 +
p
1 2
2
n n
1 2









– Unequal-variances t-interval
s2 s 2
(x – x )  t a  1 + 2
1 2
n
2 n
1
2
© 2008 Thomson South-Western
Confidence Interval for (µ1 – µ2)
• The (1 – a)% confidence interval for the
difference in two means:
– Known-variances z-interval
( x1  x2 )  za 2
s
2
1
n1
+
s
2
2
n2
© 2008 Thomson South-Western
Confidence Interval for (p1 – p2)
• The (1 – a)% confidence interval for the
difference in two proportions:
p (1– p )
p (1– p )
1 + 2
2
(p – p )  z a  1
1 2
n
n
2
1
2
– when sample sizes are sufficiently large.
© 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, x , across all data... total variation in the data
(not variance).
SST = (x – x)2
ij
© 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... betweengroups variation (not variance).
SSTR =  n (x – x)2
j j
– Sum of squares error (SSE) = sum of squared
differences between the individual data values and the
mean for the group to which each belongs... withingroup variation (not variance).
SSE =  (x – x )2
ij j
© 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... betweengroups 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:
(O – E )2
j
j
2
=
c

Ej
where Oj = Actual number observed in
each class
Ej = Expected number, pj • n
© 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:
(O – E )2
c 2 =   ij ij
E
ij
© 2008 Thomson South-Western
Chi-Square Tests of Multiple p’s
• Rejection Region:
Degrees of freedom: df = (k – 1)
• Test Statistic:
(O – E )2
ij
ij
2
c = 
E
ij
© 2008 Thomson South-Western
Determining the Least Squares
Regression Line
• Least Squares Regression Line:
yˆ = b0 + b1x1
– Slope
( x y ) – n x  y
i i
b =
1
( x 2 ) – n x 2
i
– y-intercept
b0 = y – b1x
© 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 e
– The square root of MSE given by ANOVA
( yi – yˆ )2
s y,x =
n–2
© 2008 Thomson South-Western
Equations for the Interval Estimates
• Confidence Interval for the Mean of y
2
ˆy  ta (s y,x) 1n + (x value – x)
( x )2
2
( x 2) – ni
i
• Prediction Interval for the Individual y
ˆy  ta (sy,x ) 1 + 1n + (x value – x )2
( x )2
2
i
( x 2 ) –
n
i
© 2008 Thomson South-Western
Coefficient of Correlation, r and
Coefficient of Determination, r2
r=
 x y )  ( x )( y )
n( x )  ( x ) * n( y )  ( y )
n(
2
i
i i
i
2
i
i
2
i
i
2
=0
Three Tests for Linearity
• 1. Testing the Coefficient of Correlation
H0: r = 0 There is no linear relationship between x and y.
H1: r  0 There is a linear relationship between x and y.
r
Test Statistic: t =
1 – r2
n– 2
• 2. Testing the Slope of the Regression Line
H0: b1 = 0 There is no linear relationship between x and y.
H1: b1  0 There is a linear relationship between x and y.
Test Statistic:
t=s
b
1
y,x
 x2  n( x )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.
SSR
Test Statistic: F = MSR =
1
MSE
SSE
(n – 2)
Note: At the level of simple linear regression, the global
F-test is equivalent to the t-test on b1. When we conduct
regression analysis of multiple variables, the global Ftest will take on a unique function.
© 2008 Thomson South-Western
A General Test of b1
• Testing the Slope of the Population
Regression Line Is Equal to a Specific
Value.
H0: b1 = b10
The slope of the population regression line is b10.
H1: b1  b10
The slope of the population regression line is not b10.
Test Statistic: t =
s
b –b
1 10
y, x
2 – n( x )2
x

© 2008 Thomson South-Western