Question

A new motorcycle is being developed to meet new gas mileage standards recently passed by the federal government. An engineer tested three model designs by randomly assigning the motorcycles in a blind study (the riders recorded the mileage but did not know the design of the motorcycle, only its code). The data in miles per gallon (MPG) for the three designs is listed below:

X-1	B-1	Z-1
45	55	54
62	55	52
50	54	41
65	44	40
62	52	38
50	69	39
50	58	51
58	45	64

  determine the means for the three groups and compute the sums of squares, degrees of freedom, and mean squares for these data.

What are the hypotheses to be tested here?

What is the value of F?

Using an alpha of 0.05, is the F statistic significant? Why?

What is the null and alternative hypothesis? What is your conclusion?

Answer 1

The mean of each group is,

$\mu_x=$442/8 = 55.25

$\mu_B=$ 432/8=54

$\mu_z=$379/8=47.375

Grand total = G=1253

N=24

n=8

the sum of squares for between groups = SSB= $\frac{\sum \mu_i^2}{n}-\frac{G^2}{N}$ = 286.5833

the sum of square for total=SST = $\sum x_i^2-\frac{G^2}{N}$ = 1703.958

Sum of square for within groups = SSW= SST-SSB=1703.958-286.5833= 1417.375

df for between group = no og groups-1 = 3-1 =2

df for total = N-1 = 24-1 =23

df for within = total- between = 23-2 = 21

Now, MS for within = SSW/df = 1417.375/21=67.49405

MS for between = SSB/df= 286.5833/2= 143.2917

And the F test statistic is given by,

F= MSB/MSW = 143.2917/67.49405= 2.123

F critical value at 5% level of significance = 3.4668 {from F-table}

Since, F< F critical, F statistic is not significant.

Our null hypothesis is that all the means are equal and our alternative hypothesis is that any one of the means is different.

Since F is not significant, we fail to reject the null hypothesis, all the means are equal.