A consumer research organization is attempting to determine whether there is any difference in mpg for fully loaded 22-foot trucks leased from three companies, A-Haul, Bertz, and Glyder. Five of these trucks are rented from each company. Each truck is driven with the same weight cargo over the same 200 mile route and the mpg recorded. The results of the test are: A-Haul 3.4 4.2 5.1 4.9 3.1 Bertz 5.1 2.0 8.7 6.7 6.1 Glyder 7.9 8.5 5.2 8.0 8.1 Perform the hypothesis test: H0 : 1 = 2 = 3 (Average mpg is the same for all three rental companies) H1 : Not all of the mean mpg is the same for the three companies An = 0.05 is used by the consumer research organization. Is there any difference in mean mpg? Find the F statistic value and state whether you reject the Null or fail to reject the Null hypothesis.
Try a one-way ANOVA test.
You will need to calculate Sum of Squares Within and Sum of Squares Between. You should have formulas to calculate these values.
Sum of Squares Total = Sum of Squares Within + Sum of Squares Between
To calculate Degrees of Freedom Between:
k - 1
Note: k = number of levels.
To calculate Degrees of Freedom Within:
N - k
Note: N = total number of values in all levels.
Degrees of Freedom Total = Degrees of Freedom Between + Degrees of Freedom Within
To calculate Mean Squares Between:
Sum of Squares Between divided by Degrees of Freedom Between
To calculate Mean Squares Within:
Sum of Squares Within divided by Degrees of Freedom Within
To calculate F-ratio:
Mean Squares Between divided by Mean Squares Within
You can set up a table with these values if asked to do so.
If the F-ratio exceeds the value from an F-table, then the null is rejected and you can conclude a difference.
If the F-ratio does not exceed the value from an F-table, then the null cannot be rejected and you cannot conclude a difference.
I hope this will help get you started.
To find out if there is any difference in mean mpg for fully loaded 22-foot trucks leased from three companies (A-Haul, Bertz, and Glyder), we can use a one-way analysis of variance (ANOVA) test.
Here are the steps to perform the hypothesis test:
1. Specify the hypotheses:
H0: 1 = 2 = 3 (Average mpg is the same for all three rental companies)
H1: Not all of the mean mpg is the same for the three companies
2. Set the significance level (α):
The consumer research organization has chosen α = 0.05.
3. Calculate the F statistic:
The F statistic compares the between-group variability to the within-group variability. If the between-group variability is significantly larger than the within-group variability, it suggests that there is a difference in mean values.
We can calculate the F statistic using the formula: F = (SSB / dfB) / (SSW / dfW), where
- SSB is the sum of squares between groups
- dfB is the degrees of freedom for between groups
- SSW is the sum of squares within groups
- dfW is the degrees of freedom for within groups
4. Determine the critical value:
We need to compare the calculated F statistic to the critical value from the F-distribution table with (k-1) and (N-k) degrees of freedom, where k is the number of groups (3 in this case) and N is the total sample size (15 in this case).
Since α = 0.05, we find the critical value from the F-distribution table at the 0.95 cumulative probability level.
5. Compare the calculated F statistic with the critical value:
- If the calculated F statistic is greater than the critical value, we reject the null hypothesis (H0) and conclude that there is a difference in mean mpg for the three rental companies.
- If the calculated F statistic is less than or equal to the critical value, we fail to reject the null hypothesis (H0) and conclude that there is no significant difference in mean mpg for the three rental companies.
Now let's calculate the F statistic value:
First, calculate the sum of squares between groups (SSB):
SSB = (n1 * (x1 - xbar)^2) + (n2 * (x2 - xbar)^2) + (n3 * (x3 - xbar)^2)
where,
n1 = 5 (number of samples from company A-Haul)
n2 = 5 (number of samples from company Bertz)
n3 = 5 (number of samples from company Glyder)
x1 = mean mpg for company A-Haul
x2 = mean mpg for company Bertz
x3 = mean mpg for company Glyder
xbar = overall mean mpg
For company A-Haul:
x1 = (3.4 + 4.2 + 5.1 + 4.9 + 3.1) / 5 = 4.14
For company Bertz:
x2 = (5.1 + 2.0 + 8.7 + 6.7 + 6.1) / 5 = 5.52
For company Glyder:
x3 = (7.9 + 8.5 + 5.2 + 8.0 + 8.1) / 5 = 7.54
xbar = (5.52 + 4.14 + 7.54) / 3 = 5.40
Now calculate SSB:
SSB = (5 * (4.14 - 5.40)^2) + (5 * (5.52 - 5.40)^2) + (5 * (7.54 - 5.40)^2) = 10.28
Next, calculate the sum of squares within groups (SSW):
SSW = (n1 - 1) * s1^2 + (n2 - 1) * s2^2 + (n3 - 1) * s3^2
where,
s1 = standard deviation for company A-Haul
s2 = standard deviation for company Bertz
s3 = standard deviation for company Glyder
Calculate s1:
s1 = sqrt(((3.4 - 4.14)^2 + (4.2 - 4.14)^2 + (5.1 - 4.14)^2 + (4.9 - 4.14)^2 + (3.1 - 4.14)^2) / (5 - 1))
= sqrt((1.204 + 0.0016 + 0.7921 + 0.4209 + 1.4641) / 4) = sqrt(0.9706) = 0.9852
Similarly, calculate s2 and s3:
s2 = sqrt(((5.1 - 5.52)^2 + (2.0 - 5.52)^2 + (8.7 - 5.52)^2 + (6.7 - 5.52)^2 + (6.1 - 5.52)^2) / (5 - 1))
= sqrt((0.1716 + 9.2704 + 10.0804 + 0.2304 + 0.0289) / 4) = sqrt(4.7002) = 2.1683
s3 = sqrt(((7.9 - 7.54)^2 + (8.5 - 7.54)^2 + (5.2 - 7.54)^2 + (8.0 - 7.54)^2 + (8.1 - 7.54)^2) / (5 - 1))
= sqrt((0.2076 + 0.2176 + 5.1556 + 0.2116 + 0.2164) / 4) = sqrt(1.0772) = 1.0388
Finally, calculate SSW:
SSW = (4 * (0.9852^2)) + (4 * (2.1683^2)) + (4 * (1.0388^2)) = 18.9799
Now, calculate the degrees of freedom for both between and within groups:
dfB = k - 1 = 3 - 1 = 2
dfW = N - k = 15 - 3 = 12
Let's plug these values into the formula to calculate the F statistic:
F = (SSB / dfB) / (SSW / dfW)
= (10.28 / 2) / (18.9799 / 12)
= 5.14
Next, we need to determine the critical value from the F-distribution table.
Since the degrees of freedom are dfB = 2 and dfW = 12, we need to find the critical value at the 0.95 cumulative probability level.
From the F-distribution table, the critical value for α = 0.05 and df1 = 2, df2 = 12 is approximately 3.89.
Finally, compare the calculated F statistic (5.14) to the critical value (3.89):
Conclusion:
Since the calculated F statistic (5.14) is greater than the critical value (3.89), we reject the null hypothesis (H0) and conclude that there is a significant difference in mean mpg for the three rental companies.