Question

An acoustical engineer is studying noise levels in center cities with day of the work week. She takes measurements each day of the week for four weeks at the same intersection and at the same time (5 pm). The following measurements (decibels) were taken:

M	Tu	W	Th	F
66	76	66	73	70
71	74	73	67	66
70	73	68	67	65
65	75	71	71	67

Use a 5% level.

(a) perform a Tukey multiple comparison test

(b) Use the Neuman-Keuls approach to examine the difference among means

Answer 1

To test whether there is any significant difference in the noise levels in center cities with day of the work week,

Here, day of the work week are the treatments whose effects are to be tested.The study is in the form of a CRD layout.

Using SPSS,

Conducting a One way ANOVA,

We find that, the means of the noise level for the 5 treatments (work days of the week) do differ significantly(p value = 0.01<0.05) at alpha = 0.05.

To identify the exact pairs that differ, we may perform post hoc test by making pairwise comparison of the means.

(a) Tukeys test compares, every possible pair of means, using critical value approach,

test statistic given by, T =

where, are the pair of means to be compared and MS is the mean square within and n = no. of observations in the category compared.

Compare T with the Tukey's critical tabled value for the corresponding df for within and flag the significantly different means:

From Tukey's test, we can see that, Noise levels on Tuesday differs significantly from the rest, (in particular, from Monday and Friday).But other means do not differ significantly.

(b) Neuman Keuls differs from Tukeys test in that it adopts a step wise comparision of means unlike all pairs tested as in Tukey's.

Suppose, ,

Fist comparision of means 1 and 4 are made.If there is no significant difference, the test is stopped, (since homogeneity in extreme means implies every mean within them would produce the same result).But if means of 1 and 4 do differ, then next comparision is made with the next larger range and continued till all the differences are identified.

From Neuman Keuls test, it is clearly visible that the mean noise level on Tuesday is significantly different from those in other week days.