Question

The accompanying data table shows the energy consumed​ (in millions of​ Btu) in one year for a random sample of households from four regions. At alpha equals 0.10​, can you conclude that the mean energy consumption of at least one region is different from the​ others? Perform a​ one-way ANOVA test by completing parts a through d. If​ convenient, use technology to solve the problem. Assume that each sample is normally distributed and that the population variances are equal. LOADING... Click to view the data table of regional energy consumption.

North West Midwest South West

136.1 132.1 56.5 124.8

109.5 79.3 63.2 58.3

101.5 61.6 50.6 46.2

114.6 98.6 51.3 108.4

160.3 68.8 127.7

182.0

Answer 1

Null hypotheses H0 : The average energy consumption is same for all regions.

Alternative hypotheses H1 : At least one average energy consumption is different among the four regions.

Level of significance = 0.10

Degree of freedom of group = Number of level - 1 = 4 - 1 = 3

Degree of freedom of error = Number of observations - Number of level = 20 - 4 = 16

Critical value of F at DF = 3, 16 is 2.46

Anova Summary

Source	DF	SS	MS	F
Group	3	12923.55	4307.85	4.087
Error	16	16863.75	1053.984
Total	19	29787.3

Let T_i be the total energy consumed​ for region i, n_i be number of observations of region i.

Let G be the total energy consumed​ of all observations and N be total number of observations.

ΣX² is sum of squares of all observations.

T1 = 804, T2 = 440.4 , T3 = 349.3, T4 = 337.7

G = 804 + 440.4 + 349.3 + 337.7 = 1931.4

ΣX² = 112769 + 41988.86 + 28685.83 + 32858.93 = 216302.6

SST = ΣX² - G²/N = 216302.6 - 1931.4²/20 = 29787.3

SSTR = ΣT²/n - G²/N = (804² /6 + 440.4² /5 + 349.3² /5 + 337.7² /4) - 1931.4²/20 = 12923.55

SSE = 29787.3 - 12923.55 = 16863.75

MSTR = SSTR / df for group = 12923.55 / 3 = 4307.85

MSE = SSE / df for error = 16863.75 / 16 = 1053.984

F = MSTR / MSE = 4307.85 / 1053.984 = 4.087

As, the observed value of F (4.087) is greater than the critical value (2.46), we reject the null hypothesis and conclude that the at least one average energy consumption is different among the four regions.