A dairy farmer is looking at methods for transporting milk from her farm to a da
ID: 3152956 • Letter: A
Question
A dairy farmer is looking at methods for transporting milk from her farm to a dairy plant. Three different methods are trialed over fourteen working days, and the daily costs of the methods (in $ 100) were as follows:
Part b) A table summarizing the data is given below. Provide the missing standard deviation (to three decimal places):
Part c) Complete the ANOVA table below, giving answers to three decimal places (please keep at least 6 decimal places in all intermediate steps and use the non-rounded values for all calculations):
Part d) Taking that the underlying assumptions of ANOVA hold and that the approach will be applied, what is the estimate of the common variance of the daily costs for the three delivery methods? (Give your answer to three decimal places.)
Part e) Which of the following inferences can be made when testing the null hypothesis that the three delivery methods have the same mean daily cost?
A. Since the observed F statistic is greater than F=0.05,2,11F=0.05,2,11 we can reject the null hypothesis that the three delivery methods have the same mean daily cost.
B. Since the observed F statistic is greater than F=0.05,2,11F=0.05,2,11 we do not reject the null hypothesis that the three delivery methods have the same mean daily cost.
C. Since the observed F statistic is less than F=0.05,2,11F=0.05,2,11 we do not reject the null hypothesis that the three delivery methods have the same mean daily cost.
D. Since the observed F statistic is less than F=0.05,2,11F=0.05,2,11 we can reject the null hypothesis that the three delivery methods have the same mean daily cost.
Explanation / Answer
A dairy farmer is looking at methods for transporting milk from her farm to a dairy plant. Three different methods are trialed over fourteen working days, and the daily costs of the methods (in $ 100) were as follows:
Method 1
5.51
6.47
7.08
5.22
Method 2
6.32
4.96
6.70
7.55
9.08
Method 3
10.11
9.17
8.17
7.22
8.33
Box plots displaying the data are shown below:
Part a) TRUE or FALSE: If applying the analysis of variance (ANOVA) to these data, we must assume...
i) The sample mean costs for the three methods are equal
A. True
B. False
ii) The daily costs from each method are from a Normal distribution
A. True
B. False
iii) The daily costs for each method are independent.
A. True
B. False
iv) The sample standard deviations of the costs for each method are equal.
A. True
B. False
v) The daily costs for the different methods are independent.
A. True
B. False
Part b) A table summarizing the data is given below. Provide the missing standard deviation (to three decimal places):
Method
Sample size
Mean
Standard deviation
1
4
6.070
0.860
2
5
6.922
1.526
3
5
8.600
1.092
Part c) Complete the ANOVA table below, giving answers to three decimal places (please keep at least 6 decimal places in all intermediate steps and use the non-rounded values for all calculations):
Source of Variation
df
Sum of Squares
Mean Squares
F-ratio
Treatment
2
15.209
7.605
5.133
Error
11
16.296
1.481
Total
13
31.505
Part d) Taking that the underlying assumptions of ANOVA hold and that the approach will be applied, what is the estimate of the common variance of the daily costs for the three delivery methods? (Give your answer to three decimal places.)
Method 1
Method2
Method3
5.51
6.32
10.11
6.47
4.96
9.17
7.08
6.7
8.17
5.22
7.55
7.22
9.08
8.33
deviation square from group mean
0.3136
0.362404
2.2801
0.16
3.849444
0.3249
1.0201
0.049284
0.1849
0.7225
0.394384
1.9044
4.656964
0.0729
Total =16.29588
Common variance = 16.29588/12=1.358
common variance = 1.358
Part e) Which of the following inferences can be made when testing the null hypothesis that the three delivery methods have the same mean daily cost?
A. Since the observed F statistic is greater than F=0.05,2,11F=0.05,2,11 we can reject the null hypothesis that the three delivery methods have the same mean daily cost.
B. Since the observed F statistic is greater than F=0.05,2,11F=0.05,2,11 we do not reject the null hypothesis that the three delivery methods have the same mean daily cost.
C. Since the observed F statistic is less than F=0.05,2,11F=0.05,2,11 we do not reject the null hypothesis that the three delivery methods have the same mean daily cost.
D. Since the observed F statistic is less than F=0.05,2,11F=0.05,2,11 we can reject the null hypothesis that the three delivery methods have the same mean daily cost.
Method 1
5.51
6.47
7.08
5.22
Method 2
6.32
4.96
6.70
7.55
9.08
Method 3
10.11
9.17
8.17
7.22
8.33
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