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

(a) Does the noise level differ from day-to-day? Use a 5% level.

(b) Use the Duncan test to determine which days of the week show different noise levels

Answer 1

(a)

ANOVA:

To test the difference of noise leve from day-to-day, one-way ANOVA was used.

Null hypothesis:

H0: Mean noise level dose not differ from day-to-day.

Alternative hypothesis:

H1: There is a significant difference in mean noise level in atleast one day.

Level of significance:

p-value:

ANOVA was carried out using Excel. Below are the steps to produce ANOVA ouput in Excel.

1. Enter the given data as shown below in Excel

2. Go to Data tab and Select Data Analysis

3. Select ANOVA: Single factor, Click OK

4. Select the data ranges and check-in the options as shown below

5. Click Ok. Below ouput will be created in a new sheet.

p-value for the test is 0.0096 which is less than the level of significance of 0.05 (5%). Hence, we would reject the null hypothesis and conclude that there is a significant difference in mean noise level across days.

(b)

Duncan Test:

The critical value for Duncan test is calculated as

Q_p is the value from Q-table for Error degrees of freedom and number of groups (days)

MSE is Mean sum of squares of Errors (within groups) from the ANOVA table

r is the number of observations in each day

From ANOVA table, MSE = 6.73 and Number of observations in each group, r = 4.

The duncan critical value is calculated for 2, 3, 4 and 5 groups (days). That is for p = 2,3,4,5.

With constant degrees of freedom = 15 and 5% level of significance

For p = 2, Q₂ = 3.014

For p = 3, Q₃ = 3.160

For p = 4, Q₄ = 3.250

For p = 5, Q5 = 3.312

Now, D_p for p=2,3,4 and 5 is calculated as below

Arrange means of all the groups in ascending order:

Groups	Average
F	67
M	68
W	69.5
Th	69.5
Tu	74.5

Comparing the D_p and Difference of the means

Friday Vs all other days:

	Distance between two days order + 1 = p	Dp
Difference of Friday and Monday = 68-67 = 1	2	3.9095
Difference of Friday and Wednesday = 69.5 - 67 = 2.5	3	4.0989
Difference of Friday and Thursday = 69.5 - 67 = 2.5	4	4.2156
Difference of Friday and Tuesday = 74.5 - 67 = 7.5	5	4.296

The difference of Friday and Tuesday is greater than their corresponding D_p. Hence, we conclude that there is a significant mean difference in noise level between Tuesday and Friday.

Monday Vs all other days except Friday as the comparison with Friday is already done in the above table

	Distance between two days + 1 = p	Dp
Difference of Monday and Wednesday = 69.5 - 68 = 1.5	2	3.9095
Difference of Monday and Thursday = 69.5 - 68 = 1.5	3	4.0989
Difference of Monday and Tuesday = 74.5 - 68 = 6.5	4	4.2156

The difference of Monday and Tuesday is greater than their corresponding D_p. Hence, we conclude that there is a significant mean difference in noise level between Tuesday and Monday.

Wednesday Vs all other days except Friday and Monday as the comparison with Wednesday has already been done in the above tables

	Distance between two days + 1 = p	Dp
Difference of Wednesday and Thursday = 69.5 – 69.5 = 0	2	3.9095
Difference of Wednesday and Tuesday = 74.5 – 69.5 = 5	3	4.0989

The difference of Wednesday and Tuesday is greater than their corresponding D_p. Hence, we conclude that there is a significant mean difference in noise level between Tuesday and Wednesday.

Thursday Vs Tuesday, as the comparison of Thursday Vs all other days has already been done in the above tables.

	Distance between two days + 1 = p	Dp
Difference of Thursday and Tuesday = 74.5 – 69.5 = 5	2	3.9095

The difference of Thursday and Tuesday is greater than their corresponding D_p. Hence, we conclude that there is a significant mean difference in noise level between Tuesday and Thursday.

In all the above comparisons mean noise level on Tuesdays is significanlty different from all other days at 5% level of significance.