Transcript MKT 317 February 12, 2010

MKT 317
February 12, 2010
AGENDA
 Quiz 1
 F-Test
 Analysis of Variance (ANOVA)
F – Test
 Why do we use an F – Test?
To examine the equality of variances in two
populations for a business research question
 Two Tailed Test:
 H0: σ12 = σ22
 H1: σ12 ≠ σ22
 One Tailed Test:
 H0: σ12 ≤ σ22
 H1: σ12 > σ22
*ALWAYS RIGHT TAILED*
Several Rules for F-Test
 One-tailed F-test
Always write a right-tailed hypothesis
Do not divide  by 2 when look for the critical value
 Two-tailed F-test
Use the right tail in the test
Divide  by 2 when look for the critical value
 Always:
Name the population with the larger expected variance to be
the population #1.
Reject H0 if Fcalculated > Fcritical
F – Test (cont.)
F calc =
Larger Sample Variance
Smaller Sample Variance
=
SL2
SS2
Degrees of Freedom for FCRITICAL:
dfNUMERATOR:
n of sample in numerator – 1
dfDENOMINATOR:
n of sample in denominator – 1
Example for Comparison of Variance - 1
The graduate student union is preparing for the
annual faculty versus graduate T.A. golf outing. The
president of the union is worried about which team is
more consistent (i.e. less variable in their scores). He
has the scores from 25 out of the 50 times this
challenge has happened. This is what he knows:
Faculty
Graduate T.A.s
n1 = 25
s1 = 9.2 strokes
n1 = 25
s1 = 4.5 strokes
Is there a difference in the variances of the scores?
Use =0.10
Example for Comparison of Variance - 2
The marketing manager for Chrysler wants to
test her hunch that women’s opinions of
minivan safety varies more then men’s
opinions. On a 100 point rating scale she
surveyed 41 women and 31 men. The women
had a standard deviation of 10.95 and the
men had a standard deviation of 8.94.
As her assistant you need to
test this hunch and report
the results. Use =0.05
ANOVA
 ANOVA lets you compare the means of multiple
populations with a single test simultaneously.
 All populations must be normally distributed and
have equal variances.
 Analysis of variance is a test for comparing
means, not variances.
ANOVA Business Examples
 Financial Analysts would like to know the effect
of the stock market (bull, bear, neutral market)
on short term interest rates.
 Logistics people want to know the effect of the
size of the shipping container (small, medium,
large) on the cost of shipping , or the time it
takes to ship it.
 Marketing people want to know the effect of
store location (downtown, suburban, rural) on
store sales.
 Accountants would like to know if the amount of
accounts receivable is effected by the size of
corporation (small, medium, large).
Hypotheses for ANOVA

There is only one set of hypotheses for all
ANOVA’s.
H0: 1= 2= 3=…= r (where r is the number of
populations [treatments]
being compared)
H1: “Not all population means are equal”
OR
“At least one population mean is different”
Important Notes
 The following hypothesis cannot be tested using
ANOVA:
H0: 1 2  3  …  r
 It takes only one mean being different from the rest
to reject H0.
 The alternative hypothesis does not say “all means
are different” OR
H1: 1 2  3  …  r
ANOVA Logic
 Our goal is to test whether the population
means of the treatments are equal to
each other.
 ANOVA compares the between–group
variance with the within-group variance.
If the treatment means are different, the
between-group variance should be
much larger compared to the withingroup variance.
 In fact, ANOVA simply consists of
calculating the ratio of the two and
measuring its significance.
Sum Squares
 Relationship between the three sum of squares:
SST = SSTR + SSE
SSTR
between group variance
SSE
within group variance
 To be able to compare these variances, we
need to take their averages.
 To take their averages, we divide each sum of
squares with its respective degrees of freedom.
Mean Squares
 Mean Square Treatment (MSTR):
SSTR
MSTR 
r 1
 Mean Square Error (MSE):
SSE
MSE 
N r
r: Number of treatments.
N: Total number of observations.
Conducting the Test
 Now that we have calculated the average
variances, we are ready to calculate a teststatistic.
 Since it is a test of variances, it will be an F-test:
MSTR
F(r-1, N-r) = ————
MSE
 The smaller the between-group variance (MSTR),
and thus the ratio (MSTR/MSE), the more likely
the means are equal. The larger it is, the more
likely that the means are not equal.
 This is always a right-tailed test.
 Reject when Fcalculated > Fcritical
ANOVA Table
Source of
Variation
Treatment
Error
TOTAL
Sum of
Squares
SSTR
SSE
SST
df
r-1
N-r
N-1
Mean
Square
MSTR
MSE
SSTR
MSTR 
r 1
Fcal 
SSE
MSE 
N r
Fcrit  F ;r 1, N  r
SST = SSTR + SSE
F-ratio
Fcal
MSTR
MSE
N-1 = N-r + r-1
ANOVA EXAMPLE 1
The manager of a store wants to decide what kind of
hand-knit sweaters to sell. The manager is considering
three kinds of sweaters: Irish, Peruvian, and Shetland.
The decision will depend on the results of an analysis of
which kind of sweater, if any, lasts the longest before
wearing out. There are 20 observations on Irish
sweaters, 18 on Peruvian sweaters, and 21 on Shetland
sweaters. The data are assumed to be independent
random samples from the three populations of
sweaters. The manager hires a statistician, who carries
out an ANOVA and finds SSE = 1,240 and SSTR = 740.
Construct a complete ANOVA table, and determine
whether there is evidence to conclude that the three
kinds of sweaters do not have equal average durability.
ANOVA EXAMPLE 2
Kevin Kelly is shopping for wellies on the Internet. He
has found 400 different pairs of wellies for sale from
companies in Ireland, England, China, and the U.S.
(100 from each country). He is wondering if the
average price for a pair of wellies is the same in each
country.
Here is what he knows:
SSTR = 270
MSE = 9
Construct a complete ANOVA table
and determine whether there is evidence to
conclude that the four countries do not
have equal average prices.