A pet food company conducted an experiment to compare five different cat foods.
ID: 3265508 • Letter: A
Question
A pet food company conducted an experiment to compare five different cat foods. A sample of 30 cats from the population at a local animal shelter was selected. Six cats were randomly assigned to each of the five products. Each was then presented with 3 ounces of the selected food in a dish at feeding time. The researchers defined the variable to be measured as the number of ounces of food that the cat consumed within a 10-minute time interval. The results for this experiment are summarized in the accompanying table.
Complete (a) through (d) below.
Kidney Shrimp Chicken_Liver Salmon Beef
2.39 2.26 2.28 1.78 1.74
2.64 2.69 2.24 1.84 1.52
2.31 2.26 2.42 1.95 1.18
2.46 2.44 2.68 2.05 1.64
2.59 2.35 2.26 2.27 1.68
2.62 2.38 2.57 1.59 1.32
Critical values of the Studentized Range, Q. Upper 5% points (alpha=0.05).
Numerator, df
Denominator, df 2 3 4 5 6 7 8 9 10
1 17.97 26.98 32.82 37.08 40.41 43.12 45.40 47.36 49.07
2 6.09 8.33 9.80 10.88 11.74 12.44 13.03 13.54 13.99
3 4.50 5.91 6.83 7.50 8.04 8.48 8.85 9.18 9.46
4 3.93 5.04 5.76 6.29 6.71 7.05 7.35 7.60 7.83
5 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 7.00
6 3.46 4.34 4.90 5.31 5.63 5.90 6.12 6.32 6.49
7 3.34 4.17 4.68 5.06 5.36 5.61 5.82 6.00 6.16
8 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92
9 3.20 3.95 4.42 4.76 5.02 5.24 5.43 5.60 5.74
10 3.15 3.88 4.33 4.65 4.91 5.12 5.31 5.46 5.60
11 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49
12 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.40
13 3.06 3.74 4.15 4.45 4.69 4.89 5.05 5.19 5.32
14 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25
15 3.01 3.67 4.08 4.37 4.60 4.78 4.94 5.08 5.20
16 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15
17 2.98 3.63 4.02 4.30 4.52 4.71 4.86 4.99 5.11
18 2.97 3.61 4.00 4.28 4.50 4.67 4.82 4.96 5.07
19 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04
20 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01
24 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92
30 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82
40 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.64 4.74
60 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65
120 2.80 3.36 3.69 3.92 4.10 4.24 4.36 4.47 4.56
infinity 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47
a. At the 0.05 level of significance, is there evidence of a difference in the mean amount of food eaten among the various products?
Determine the hypotheses. Choose the correct answer below.
A. H0: 1 = 2 = • • • = 5
H1: Not all j are equal (where j =1,2, ...,5)
B. H0: 1 = 2 = • • • = 6
H1: Not all j are equal (where j =1,2,...,6)
C. H0: 1 = 2 = • • • =5
H1: 1 2 • • • 5
D. H0: 1 = 2 = • • • = 6
H1: 1 2 • • • 6
Find the test statistic.
FSTAT = ____?(Round to two decimal places as needed.)
Determine the critical value.
F = ____? (Round to two decimal places as needed.)
Reach a decision.
(Reject, Do not reject) H0. There is (insufficient, sufficient) evidence of a difference in the mean amount of food eaten among the various products.
b. If appropriate, determine which products differ in the mean amount of food eaten.
Let the kidney-based cat food be group 1, the shrimp-based cat food be group 2, the chicken liver-based cat food be group 3, the salmon-based cat food be group 4, and the beef-based cat food be group 5.
Is there significant evidence that group 1 and group 2 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 1 and group 3 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 1 and group 4 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 1 and group 5 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 2 and group 3 differ in the mean amount of food eaten?
A. No
B. Yes
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 2 and group 4 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 2 and group 5 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 3 and group 4 differ in the mean amount of food eaten?
A. No
B. Yes
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 3 and group 5 differ in the mean amount of food eaten?
A. Yes
B. No
C. It is not appropriate to test which products differ in the mean amount of food eaten.
Is there significant evidence that group 4 and group 5 differ in the mean amount of food eaten?
A. No
B. Yes
C. It is not appropriate to test which products differ in the mean amount of food eaten.
c. At the 0.05 level of significance, is there evidence of a difference in the variation in the amount of food eaten among the various products?
Determine the hypotheses. Choose the correct answer below.
A. H0: 1 = 2 = • • • = 5
H1: Not all j are equal (where j = 1,2,...,5)
B. H0: 2 = 2 = • • • = 6
H1: Not all j are equal (where j =1,2, ...,6)
C. H0: 1 = 2 = • • • = 5
H1: 1 2 • • • 5
D. H0: 1 = 2 = • • • = 6
H1: 1 2 • • • 6
Find the test statistic.
FSTAT = ____?(Round to two decimal places as needed.)
Determine the critical value.
F = ____? (Round to two decimal places as needed.)
Reach a decision.
(Reject, Do not reject) H0. There is (sufficient, insufficient) evidence to conclude that there is a difference in the variation in the amount of food eaten among the various products.
d. The pet food company has a business objective of expanding its product line beyond its current kidney- and shrimp-based cat foods. The company developed two new products, which are the chicken liver- and salmon-based cat foods. The beef-based cat food is a generic product sold in a supermarket chain. What should the company conclude in terms of expanding its product line?
Choose the correct conclusion below.
A. The chicken liver-based cat food is the best choice for expansion. The salmon-based cat-food is a possibility for expansion; its mean amount of food eaten is significantly larger than the beef-based cat food, but smaller than the other cat foods.
B. The salmon-based cat food is the best choice for expansion. The chicken liver-based cat-food is a possibility for expansion; its mean amount of food eaten is significantly larger than the beef-based cat food, but smaller than the other cat foods.
C. It does not matter which product(s) is/are used for expansion because there are no significant differences in the mean amounts eaten among the various products.
Explanation / Answer
A pet food company conducted an experiment to compare five different cat foods. A sample of 30 cats from the population at a local animal shelter was selected. Six cats were randomly assigned to each of the five products. Each was then presented with 3 ounces of the selected food in a dish at feeding time. The researchers defined the variable to be measured as the number of ounces of food that the cat consumed within a 10-minute time interval. The results for this experiment are summarized in the accompanying table.
Here we have given five groups and we have to test five means. For testing more than two means we use one way anova.
The hypothesis of the test is,
H0 : mu1 = mu2 = mu3= mu4 = mu5
H1 : Atleast one of the mean is differ than 0.
or
H1: Not all j are equal (where j =1,2, ...,5)
Assume alpha = level of significance = 0.05
In one way Anova the test statistis follows F-distribution.
Input data on k=5 independent treatments:
Descriptive statistics for k=5 independent treatments:
One-way ANOVA of your k=5 independent treatments:
Conclusion from Anova:
The p-value corresponing to the F-statistic of one-way ANOVA is lower than 0.05, suggesting that the one or more treatments are significantly different.
Therefore we use Scheffé test to check which treatmen differ significantly.
Scheffé multiple comparison :
We define a statistic named T as the ratio of unsigned contrast mean to contrast standard error.
It can be show that for contrasts that are treatment pairs (i,j) with unit coefficents,
Ti,j = Qi,j / sqrt(2)
where Qi,j is the Q-statistic that was created for the Tukey HSD test.
Scheffe's method provides a formula which directly leads to the Scheffé p-value corresponding to an observed value of T as:
1 - F(T^2 / k-1, k-1, v)
where F() is the cumulative F distribution with its two degrees of freedom parameters k1 and . Note that k is the number of treatments and is the degrees of freedom of error that were established earlier.
The Scheffé p-value of the observed T-statistic Ti,j is shown below for all relevant pairs of treatments, along with color coded Scheffé inference (red for insignificant, green for significant) based on the p-value.
We see that pairs with green colour indicates that the pairs are significant at 5% level of significance.
Pair numbers (1,4), (1,5),(2,4),(2,5),(3,4),(3,5) and (4,5) are significant.
Is there significant evidence that group 1 and group 2 differ in the mean amount of food eaten?
No
Is there significant evidence that group 1 and group 3 differ in the mean amount of food eaten?
No
Is there significant evidence that group 1 and group 4 differ in the mean amount of food eaten?
yes
Is there significant evidence that group 1 and group 5 differ in the mean amount of food eaten?
yes
Is there significant evidence that group 2 and group 3 differ in the mean amount of food eaten?
No
Is there significant evidence that group 2 and group 4 differ in the mean amount of food eaten?
yes
Is there significant evidence that group 2 and group 5 differ in the mean amount of food eaten?
yes
Is there significant evidence that group 3 and group 4 differ in the mean amount of food eaten?
yes
Is there significant evidence that group 3 and group 5 differ in the mean amount of food eaten?
Yes
Is there significant evidence that group 4 and group 5 differ in the mean amount of food eaten?
Yes
Treatment A B C D E Input Data 2.392.64
2.31
2.46
2.59
2.62 2.26
2.69
2.26
2.44
2.35
2.38 2.28
2.24
2.42
2.68
2.26
2.57 1.78
1.84
1.95
2.05
2.27
1.59 1.74
1.52
1.18
1.64
1.68
1.32
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