Question

Gasoline mileage (mpg) was measured on several cars of each of four different makes (coded 1, 2, 3 and 4). The make of each car is stored in the first column, and the mileage for each car is stored in the second column, of Table A. You need to conduct an analysis of variance to see if there are differences among the four makes in gasoline mileage. You should also estimate the mileage of each of the four makes of cars.
1. What is the value of the F‑statistic for testing the null hypothesis that there are no differences in gasoline mileage among the four makes of automobile?
2. What are the degrees of freedom associated with the numerator of this test statistic?
3. What are the degrees of freedom associated with the denominator of the F‑value for MAKE of car?
4. What is the estimate of the pooled variance within makes of cars (i.e. the Error mean square)?
5. What are the degrees of freedom for this variance in #4?

1	20
1	16.8
1	24.1
1	12.7
1	10.2
1	18.7
1	18.8
1	14.7
2	9.2
2	7.1
2	14.3
2	13.5
2	12.2
2	10.3
2	9.2
3	15.5
3	22.8
3	18.5
3	24
3	25.3
3	23
3	15.9
3	23
3	21.2
4	7
4	13.3
4	11.4
4	15.7
4	20.8
4	14.1
4	19.2
4	16.1

Answer 1

Solution: We can use the excel ANOVA: Single Factor data analysis tool to find the answer to the given questions. The excel steps are:

Enter the data in excel as:

Click on Data > Data Analysis > ANOVA: Single Factor > Ok

Input Range: Select all the data range including the labels.

Mark the labels in the first row

Alpha: 0.05

Choose the output cell

The excel output is given below:

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
1	8	136	17	19.3714
2	7	75.8	10.8286	6.7590
3	9	189.2	21.0222	12.6594
4	8	117.6	14.7	18.9886
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	434.3689	3	144.7896	9.8796	0.0001	2.9467
Within Groups	410.3498	28	14.6554
Total	844.7188	31

1. What is the value of the F‑statistic for testing the null hypothesis that there are no differences in gasoline mileage among the four makes of automobiles?

Answer:
2. What are the degrees of freedom associated with the numerator of this test statistic?

Answer:
3. What are the degrees of freedom associated with the denominator of the F‑value for MAKE of car?

Answer:
4. What is the estimate of the pooled variance within makes of cars (i.e. the Error mean square)?

5. What are the degrees of freedom for this variance in #4?