Question

1. Penn State OPP is preparing for the upcoming spring season after a long dragging winter. Their current mowers are overdue, and the purchasing department is ready to invest in new equipment. The manufacturer claims that their lawn mower can run for 2 hours on a full tank. OPP borrows six lawnmowers of the same make and model to test the claim in order to plan their available resources accordingly. During the study, each operator would have to make sure their tank is full of gas and would log the run time until the tank is empty. Four data points were randomly selected from each of the six mowers. The observations (in minutes) are shown in the data below:

Mower
1	2	3	4	5	6
119.48	121.73	120.79	121.84	121.01	120.19
120.75	121.54	119.93	122.18	119.29	119.81
120.26	121.20	121.20	121.95	120.41	121.05
120.68	121.76	121.69	121.84	120.23	121.76

Do the following calculations by hand:

(a) State the hypotheses that you would be interested in testing in this case.

(b) Construct the ANOVA table.

(c) Is there any significant difference in the run time due to the lawn mower? Use α = 0.05.

(d) Use Tukey’s test to compare all possible pairs of means. Use α = 0.05.

Answer 1

a) Null Hypothesis: The mean run time until the tank is empty for all 6 mowers is the same.

Alternative hypothesis: At least on mower has different mean run time until the tank is empty.

b)

ANOVA table:

Source	DF	SS	MS	F	P
Mower	5	9.522	1.904	5.05	0.005
Error	18	6.786	0.377
Total	23	16.308

S = 0.6140 R-Sq = 58.39% R-Sq(adj) = 46.83%

(c) Is there any significant difference in the run time due to the lawn mower? Use α = 0.05.

Ans: The estimated p-value for F-test statistic is 0.005 and less than 0.05 significance level. Hence, reject the null hypothesis and conclude that there is any significant difference in the run time due to the lawn mower at the 0.05 significance level.

(d) Use Tukey’s test to compare all possible pairs of means. Use α = 0.05.

Ans: Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Mower

From the above confidence intervals, if the confidence interval for ith and jth mowers include the value zero in the confidence interval, the corresponding two population means are not significantly different at 0.05 significance level.

So, we can conclude that run time due to the lawn mower is different between the 1 & 4 and 4 & 5.