STA 106 Winter Quarter, 2021 Exam II
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STA 106
Winter Quarter, 2021
Exam II
(10 points) True/False
For each of the following questions indicate true or false, then fully explain your answer. You may use examples to illustrate
your answer.
(I) (2.5 points) If a significant factor B affect is present where b = 1, we would expect all groups associated with factor B to significantly effect the mean.
(II) (2.5 points) In a Two Factor ANOVA problem, γi represents a constant.
(III) (2.5 points) If a significant interaction present, we would expect the lines drawn on an interaction plot to be approxi- mately parallel.
(IV) (2.5 points) A Tukey multiplier can be used to create pairwise confidence intervals.
Full Detail
Work out the following problems. Show your work.
1. (14 points) A study was conducted which examined the carapace length (in mm) of two species of crab (Factor A, with levels B = Blue, O = Orange), and the gender of the crab (Factor B, M = Male, F = Female). The SSE values for various models follow.
AB |
(A+B) |
A |
B |
Empty/Null |
SSE 11231.82 |
11686.94 |
11755.27 |
12263.24 |
12331.57 |
In addition, there were 200 crabs measured total.
(a) (4 points) State the “full” and “reduced” model and the associated degrees of freedom for SSE for each model, if you are testing for Factor A effects comparing to the model with no-interactions.
(b) (4 points) State the appropriate null and alternative for testing for Factor B effects (in mathematical terms), and find the appropriate test-statistic.
(c) (4 points) If we conclude that there is a significant Factor B effect, how could we answer the question “How important was Factor B to our model?” Calculate a numeric value, and explain how it answers the question.
(d) (2 points) Could we conclude that every level of Factor B has a significant effect on the outcome of carapace length? Explain your answer.
2. (14 points) Continuing with the previous problem, which examined the carapace length (in mm) of two species of crab (Factor A, with levels B = Blue, O = Orange), and the gender of the crab (Factor B, M = Male, F = Female). The table of treatment means and factor level means follows:
Further, assume nij = 50 for all i, j .
ij ●
|
F (j = 1) M (j = 2) |
i ●●
|
|
B (i = 1) O (i = 2) |
28.102 34.618 |
32.014 33.688 |
30.058 34.153 |
●j● |
31.360 32.851 |
|
(a) (4 points) If the 95% confidence interval for µ ●1 − µ ●2 is: (-2.9695497 0.6315497), does this suggest a significant
factor B effect? Explain your answer.
(b) (4 points) Calculate the 95% confidence interval for µ 11 − µ 12 . Assume the appropriate multiplier is : 1.653, and that you will use the MSE based on the (A + B) model.
(c) (3 points) Interpret your interval from (b) in terms of the problem.
(d) (3 points) Do your intervals in (a), (b) suggest a significant interaction effect? Explain your answer.
3. (12 points) Continuing with the previous problem, which examined the carapace length (in mm) of two species of crab (Factor A, with levels B = Blue, O = Orange), and the gender of the crab (Factor B, M = Male, F = Female).
(a) (3 points) Write down the “regression” notation for the “true” interaction model, being sure to specify all indicator variables.
(b) (3 points) Name one plot we could make, or test we could conduct for diagnostics, and what assumption of ANOVA it assesses.
(c) (3 points) What would the be purpose of transforming our data? That is, what may lead us to believe we may need a transformation of our data? Explain your answer.
(d) (3 points) If someone told you that they found 4 potential outliers in this data set, would you recommend removing them? Why or why not.
2023-02-19