handout_week 14
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University of Illinois, Urbana Champaign *
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Statistics
Date
Jan 9, 2024
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docx
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Lab 14: Lack of Fit Tests and Correlation
REVIEW QUESTIONS
Writing complete factorial models and drawing interactions 1)
a.) How do you write a two-factor complete factorial in a CRD? b.)
How about a three factor? c.)
What if I changed the design to complete factorial in a RCBD? d.)
What are the assumptions for a RCBD?
2.) Draw a significant three-factor interaction if A = 2 levels, B = 2 levels, and C = 3 levels. Marginal means and model selection 3) What are the kinds of marginal means we have discussed in the course thus far? How do we calculate them?
1
4) If I am reading a word problem, what clues am I looking for to know if I should be using a CRD, RCBD, or Latin square design?
4b) What are some ways we have discussed to quantify if a RCBD is better than a CRD? How do
calculate it and what does it mean?
Understanding differences in designs and fixed versus random effects
5) Compare and contrast between a fixed effect and a random effect. Give an example of an RCBD where the blocks should be fixed, and one example of an RCBD where the blocks should be random. 2
6) Suppose that an RCBD has been implemented, and that the block effect was significant in the corresponding statistical model. Moreover, suppose that the MSE from this RCBD statistical model was compared to the MSE of a CRD model. Which model would you expect to have a smaller MSE? MORE ON LINEAR REGRESSION
Lack of Fit Tests
7.) We have a dataset that takes a sample of 10 Pittsburgh Penguins players and their penalty infraction minutes (x variable) and their points scored (Y variable) during the 2017 Stanley Cup Playoffs. The data are presented below. Xi
Yij
Ybar i (Yij - Ybar i)^2
70
33
19
196
70
5
19
196
35
9
10.5
2.25
35
12
10.5
2.25
25
21
32.5
132.25
25
44
32.5
132.25
21
23
23
0
21
23
23
0
10
16
15.5
0.25
10
15
15.5
0.25
SUM: 661.5
3
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a.) Conduct a lack of fit test for these data at alpha = 0.05. Please complete the table, state the null and alternative hypotheses for the lack of fit F-test, and make a decision and conclusion. Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
F Calculated
F Critical
Model
Error
Lack of Fit
Pure Error
Corrected
Total
b.) Using the table that you have already filled out above, please determine if there is a linear relationship between X and Y at alpha = 0.05.
Correlation
8.) If the correlation coefficient is 0.8, what do we know about the linear relationship? 9.) If we do not have a very large sample size and the correlation coefficient is 0.8, what do we know about the bias? Why? 4
10.) What happens to the bias if we increase the sample size? 11.) What happens to the bias if we decrease the sample size? 12.) How are correlation coefficient and the coefficient of determination related mathematically?
Linear regression problem for independent practice
13.) A runner is interested in the relationship between the number of miles ran (the independent, or X, variable) and the number of calories burned (the dependent, or Y, variable) as a result. In particular, he is interested in the assessing the linear relationship between these two variables. Therefore, the number of calories burned from three runners running 5 miles, three runners running 10 miles, three running 15 miles, and three running 20 miles were measured, and a linear regression model was fitted to these data (a
partially filled ANOVA table is included below). Furthermore, the following table was developed to conduct a lack of fit F-test. Please use the information in this table to complete the ANOVA table and conduct a lack of fit F test.
X
i
Y
ij
Y
i
(
Y
ij
−
Y
¿¿
i
)
2
¿
5
78.20
79.08
0.78
5
76.79
5.26
5
82.26
10.09
10
313.12
313.71
0.35
10
309.69
16.16
10
318.32
21.25
5
15
701.57
700.67
0.80
15
701.75
1.16
15
698.70
3.89
20
1233.38
1235.67
5.26
20
1234.84
0.69
20
1238.80
9.78
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
F Calculated
F Critical
Model
2231228
4.96
Error
Lack of Fit
4.46
Pure Error
Corrected
Total
11
2298966
Please complete the table, state the null and alternative hypotheses for the lack of fit F-test, and make a decision and conclusion about this test. 6
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Related Questions
In the context of a Factorial Design, what is an interaction? What is an example of an interaction?
arrow_forward
A mail-order catalog firm would like to test the effect of the size of a magazine advertisement and the advertisement design on the number of catalog requests received (data in thousands). Three advertising designs and two different size advertisements were considered. The data obtained follow.
At the 0.05 level of significance,
a. Is there an interaction between type of design and size of advertisement?
b. Is there an effect due to type of design?
c. Is there an effect due to size of advertisement?
arrow_forward
A study examined the interactions in a children's museum. (Interactions by visitors to the museum included show-and-tell, learning, teaching, refocusing, participatory play, advocating, and disciplining
interactions.) Researchers observed a sample of 180 meaningful interactions, of which 86 were led by children and 94 were led by adult caregivers. Complete parts a through d below.
a. Give a point estimate of the true proportion of all meaningful interactions in a children's museum that are led by children.
The point estimate is.
(Type an integer or decimal rounded to three decimal places as needed.)
arrow_forward
A correlational study is designed to
A) test causal inferences
B) assess the assocation between two measures
C)compare the means of three or more groups
D) manipulate the independent variables
arrow_forward
]14. Many animals, including humans, tend
to avoid direct eye contact and even patterns
that look like eyes. Some insects, including
moths, have evolved eye-spot patterns on
their wings to help ward off predators.
Scaife (1976) reports a study examining how
eye-spot patterns affect the behaviour of
birds. In the study, the birds were tested in a
box with two chambers and were free to
move from one chamber to another. In one
chamber, two large eye-spots were painted
on one wall. The other chamber had plain
walls. The researcher recorded the amount
of time each bird spent in the plain chamber
during a 60-minute session. Suppose the
study produced a mean of M
34.5 minutes
on the plain chamber with SS = 210 for a
sample
of n =
15 birds. (Now: If the eye spots have
no effect. then the birds should spend an
average of u = 30 minutes in each chamber.)
a. Is this sample sufficient to conclude that
the eyes pots have a significant influence on
the bird's behaviour? Ike a two-tailed test
with…
arrow_forward
Why are interaction effects only associated with factorial designs?
Examine the following data table and indicate the presence of (a) main effect of A, (b)
main effect of B, (c) and interaction effect. Verify this with a graph.
Factor B
B1
B2
Factor A
A1
10
5
A2
5
10
arrow_forward
I need help with 1, 2 and 3 plz.
arrow_forward
A market researcher is trying to determine how age, gender, and race/ethnicity are related to whether or not people own a device for streaming content to a television. In this study, ownership of a streaming device is the:
Group of answer choices
a). dependent variable
b). causal variable
c). independent variable
d). control variable
arrow_forward
I need help with 4 and 7
arrow_forward
A two-factor research study is used to evaluate the effectiveness of a new blood-pressure medication. In this two-factor study, factor A is medication versus no medication and factor B is male versus female. The medicine is expected to reduce blood pressure for both males and females, but it is expected to have a much greater effect for males. What pattern of results should be obtained if the medication works as predicted?
A)There is no interaction
B)There is an interaction
C)More information is needed to determine about the interaction
arrow_forward
The correlation coefficient, R, measures
a) whether or not a scatterplot shows an interesting pattern.
b) whether or not a cause-and-effect relation exists between two variables.
c) the strength and direction of the linear relation between two variables.
arrow_forward
Many animals, including humans, tend to avoid direct eye contact and even pattern that look like eyes. Some insects, including months, have involved eye-spot pattern in their wings to help ward off predators. Scalfe (1976)reports a study examining how eyes spot patterns effect the behavior of birds. In the study birds were tested in a box with two chambers and were free to move from one chamber to another. In one chamber, two large eyes spot were painted on one wall. The other chamber to another. The researcher recorded the amount of time each bird spend in the plain chamber during a 60-minutes session. Suppose the study produced a mean of M=34.5 minutes in the plain chamber with SS=210 for a sample of n=15 birds ( Note if the eyes spots have no effect, then the birds should spend and average of µ=30 minutes in each chamber.)
A) Is this sample sufficient to conclude that the eyes -spot have significant influence on the birds behaviors? Use a two- Tailed test with α=.05
Compute the…
arrow_forward
In an
the researcher interacts and influences the behavior of the subjects. In an
Observational Study; Observational Study
Designed Experiment; Designed Experiment
Designed Experiment; Observational Study
Observational Study; Designed Experiment
the researcher observes the subjects and makes no effort to interact or influence the subjects.
arrow_forward
The correlation coefficient between factor score and the variables included in the study is known as:
a.
Component Matrix
b.
Rotated Component Matrix
c.
Communalities
arrow_forward
Many animals, including humans, tend to avoid direct eye contact and even patterns that look like eyes. Some insects, including moths, have evolved eye-spot patterns on their wings to help ward off predators. Scalfe (1976) reports a study examining how eye-spot patterns affect the behavior of birds. In the study, the birds were tested in a box with two chambers and were free to move from one chamber to another. In one chamber, two large eye-spots were painted on one wall. The other chamber had plain walls. The researcher recorded the amount of time each bird spent in the plain chamber during a 60-minute session. Suppose the study produced a mean of M = 34.5 minutes in the plain chamber with SS = 210 for a sample of n = 15 birds. (Note: If the eye spots have no effect, then the birds should spend an average of µ = 30 minutes in each chamber. Construct the 90% confidence interval to estimate the mean amount of time spent on the plain side for the population of birds.
arrow_forward
A study that tells us whether two variables are associated, but does not tell us if one variable causes or influences the other, is:
Question 2 options:
A)
a n experimental study without a control group.
B)
a descriptive study.
C)
a correlational study.
D)
an experimental study with one or more control groups.
arrow_forward
Factorial designs allow us to study both (blank) effects of the independent variables on the dependent variables.
A. Main and interactive
B. Symbiotic and dichotomous
C. Dependent and independent
D. Rank order and correlational
arrow_forward
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