Question

An employee of a small software company in Minneapolis bikes to work during the summer months. He can travel to work using one of three routes and wonders whether the average commute times (in minutes) differ between the three routes. He obtains the following data after traveling each route for one week.

Route 1	32	35	33	28	35
Route 2	22	24	25	24	22
Route 3	29	30	20	20	27

a-1. Construct an ANOVA table. (Round "Sum Sq" to 1 decimal place, "Mean Sq" and "F value" to 2, and round the "p-value" to 4 decimal places.)

ANOVA

Source of Variation	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Route
Residuals

a-2. At the 5% significance level, do the average commute times differ significantly between the three routes. Assume that commute times are normally distributed.

• Yes, since the p-value is less than significance level.

• Yes, since the p-value is not less than significance level.

• No, since the p-value is less than significance level.

• No, since the p-value is not less than significance level.

b. Use Tukey’s HSD method at the 5% significance level to determine which routes' average times differ. (Round difference to 1 decimal place, confidence interval bounds to 2 decimal places, and p-values to 3.)

Population Mean Difference	diff	lwr	upr	p adj	do the average times differ?
Route 2 - Route 1
Route 3 - Route 1
Route 3 - Route 2

Answer 1

a-1)

Following table shows the calculations:

	Route 1, G1	Route 2, G2	Route 3, G3
	32	22	29
	35	24	30
	33	25	20
	28	24	20
	35	22	27
Total	163	117	126

And

	G	G^2
	32	1024
	35	1225
	33	1089
	28	784
	35	1225
	22	484
	24	576
	25	625
	24	576
	22	484
	29	841
	30	900
	20	400
	20	400
	27	729
Total	406	11362

So we have

Now

So,

Since there are 5 different groups so we have k=5. Therefore degree of freedoms are:

-------------

Now

F test statistics is

So p-value of the test using excel function "=FDIST(10.55,2,12)" is 0.0023.

a-2)

Since P-value is less than 0.05 so we reject the null hypothesis. That is on the basis of sample evidence we can conclude that populations are different.
Correct option:

Yes, since the p-value is less than significance level.

b)

Here we have 3 groups and total number of observations are 15. So degree of freedom is

df= 15-3 = 12

Critical value for , df=12 and k=3 is

So Tukey's HSD will be

The sample means are

The required confidence intervals are:

groups (i-j)	xbari	xbarj	ni	nj	HSD	xbari-xbarj	Lower limit	Upper limit	Significant(Yes/No)
mu1-mu2	32.6	23.4	5	5	5.66	9.2	3.54	14.86	Yes
mu1-mu3	32.6	25.2	5	5	5.66	7.4	1.74	13.06	Yes
mu2-mu3	23.4	25.2	5	5	5.66	-1.8	-7.46	3.86	No

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Following is the screen shot of R script:

Following is the output: