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WEEK 3 Problems
Chapters 10-11
Packet
Pages:
Chapters:
Ch. 10: Between-Subjects, One-Way Analysis of Variance
Ch. 11: One-Way, Repeated Measures ANOVA
Refer to Resource Page on CANVAS for:
StatSheets
Table 4, Critical Values of F
Table 5, Studentized Range (q) Values
Online calculator/statistical app links & tutorials
Week 3 CH 10-11 Problems
1-10
11-13
Chapter 10
Between-Subjects, One-Way ANOVA
Ch. 10- Defining Key Terms
Provide definitions for the following key terms:
ANALYSIS OF VARIANCE (ANOVA):
BETWEEN-SUBJECTS, ONE-WAY ANOVA:
FACTOR:
LEVEL:
POST-HOC TEST:
SUM OF SQUARES BETWEEN (SSBetween):
SUM OF SQUARES TOTAL (SSTotal):
SUM OF SQUARES WITHIN (SSWithin):
TREATMENT EFFECT:
Week 3 CH 10-11 Problems
Page 1 of 13
Ch. 10- Identifying Levels of an Independent Variable (or â€œFactorâ€)
For each of the following research projects, please indicate the independent and dependent
variable and then specify the levels of the independent variable. (see textbook pp. 329-30 &
Stat Sheet Ch. 10).
Example:
A researcher wants to know if different color cars get hotter in the sunlight. To examine this, samples of
black cars, red cars, & white cars are left in the sunlight for 3 hours on a 70-degree day. Temperature (in
Fahrenheit) of inside cabins measured directly above the steering wheel were compared.
Independent Variable = Color of car
Levels = 1. Black, 2. Red, 3. White
Dependent Variable = Temperature (in Fahrenheit)
– – – – – – – – – – – — – – – – – – – – — – – – – – – – – — – – – – – – – – — – – – – – – – – — – – – – – – – – — – – – – – – – – RESEARCH PROJECT:
1. You want to know how color influences taste perception. Each group of participants is given
colored â€œtea,â€ which is actually just colored plain water. One group has cups with red fluid, a
second group has brown fluid, and the third group has orange fluid. The participants are to
describe the flavor of each tea and rate how well they liked each tea.
2. You want to know if music can influence heart rate. Some participants sit in a quiet room,
others hear loud heavy-metal music, while others hear soothing classical music, and another
group hears upbeat pop music. All participants have their heart rate checked after five
minutes.
3. Ryan, Hatfield, and Hofstetter investigated the effects of caffeine on memory in older adults.
All participants were between the ages of 70 and 80. One group of participants was given
decaf coffee, one group was given half-caf coffee, and one group was given full-strength
caffeinated coffee. After finishing their coffee, participants were given 16 words to memorize.
They then took both a free recall and a recognition test on the words.
Week 3 CH 10-11 Problems
Page 2 of 13
Ch. 10- Looking Up Critical Values of F (Fcv) in Table 4 & Marking the
Rare & Common Zones
For each of the following studies, look up the critical value of F (Table 4) that separates the rare zone
from the common zone & mark on the sampling distribution of the F ratio. (see textbook pp. 340-343
& Stat Sheet Ch. 10). There is also an on-line calculator for finding the critical values.
Note: ANOVAs are always 2-tailed tests with both â€œtailsâ€ wrapped into the one side.
As the effect of the treatment grows, the
numerator becomes larger than the
denominator, & the F ratio climbs above
1. As the F ratio increases, as it climbs
higher than 1, the results are more likely
to be statistically significant.
Example A:
Find the critical value of F & mark on the sampling distribution:
If dfBetween = 3 and dfWithin = 12, and alpha is set at Î± = .05?
Step 1: Go to Table 4a, Critical Value of F, alpha=.05
Step 2: Find the intersection of â€œNumeratorâ€=3 & â€œDenominatorâ€ = 12. Critical Value = 3.490
Step 3: Mark critical value on distribution.
Common
Zone
Rare Zone
Week 3 CH 10-11 Problems
Page 3 of 13
Find the critical value of F & mark on the sampling distribution:
1. If dfBetween = 3 and dfWithin = 36, and alpha is set at Î± = .05?
2. If dfWithin = 44 and dfBetween = 2, and alpha is set at Î± = .01?
3. If N = 50, and k = 4, and alpha is set at Î± = .05?
HINT: For #3, use Equation
10.1 to determine degrees of
freedom (df), then use Table 4
to find the critical value of F &
mark on the distribution.
Week 3 CH 10-11 Problems
Page 4 of 13
EXAMPLE A.
Computing a Between-Subjects One-Way ANOVA
Study: A consumer researcher gave consumers a sample shampoo. After using the shampoo, each
consumer used a scale to rate is or her satisfaction with it. Scores could range from 0 to 100, with higher
scores indicating greater satisfaction. Consumers were randomly assigned to 3 groups: 1. Store brand
of shampoo in bottle labeled as such, 2. Premium brand of shampoo in premium brand bottle, & 3. Store
brand shampoo in premium brand bottle.
A.
B.
C.
QUESTIONS:
a.
HYPOTHESES: List the Null & Alternative Hypotheses:
b.
Null Hyp: The average satisfaction ratings will be equal for the 3 treatment groups.
Alt Hyp: At least one of the average satisfaction ratings will be different from at least one other.
SET THE DECISION RULE: Using the decision rule of alpha=.05, find the degrees of freedom for
this study (Equation 10.1), & find the critical values of F (using Table 4). Use the critical values to
mark the common & rare zones on the distribution below.
K=3 groups
N=12
dfBetween = k-1 = 2
dfWithin = N-k = 12-3 = 9
Critical Value of F = 4.256
Common Zone
FCV= 4.256
Rare Zone
c.
CALCULATE THE STATISTIC: What is the calculated or computed F-test statistic for this problem?
(Equations 10.2-10.7; on-line calculator).
The statistic was calculated using Ch. 10 Calculator #1 found on the WEEK 5 RESOURCE PAGE.
The Calculated F Ratio = 41.99 (rounded to 42.00)
Week 3 CH 10-11 Problems
Page 5 of 13
INTERPRET THE RESULTS
d.
(see pp. 355+)
WAS THE NULL HYPOTHESIS REJECTED?: Is the overall ANOVA result statistically significant?
Yes, reject the Null Hypothesis, the overall ANOVA is significant. The calculated F-value of 42.00 is in the
rare zone, beyond the Critical Value of 4.256 set for the decision rule at alpha=.05 level. Conclude that at
least one of treatment group satisfaction rating is significantly different from at least one other group.
The output from Ch. 10 Calculator
#1 shows the calculated F statistic
= 42.00 with a proportion in the
tail of