We would like to conduct an analysis of variance at the 10% level of significanc

Question

We would like to conduct an analysis of variance at the 10% level of significance to compare the residential energy consumption of households in four regions of the U.S. We measure the energy consumption in a sample of 12 homes in the Northeast Region (NE), 9 homes in the Northwest Region (NW), 16 homes in the Southeast Region (SE) and 17 homes in the Southwest Region (SW).

Suppose energy consumption are known to follow a normal distribution with common standard deviation for each of the four regions.

The ANOVA table (with some values missing) is shown below:

(a) What are the hypotheses for the appropriate test of significance?

(b) Find all missing values in the table above. Show your calculations and display the final complete ANOVA table.

(c) What is the P-value for the appropriate test of significance?

(d) What is the correct conclusion of the test?

(e) Suppose you had used the critical value method to conduct the test. What would be the decision rule and the conclusion?

(f) What is the estimate of the common population standard deviation ?   

(g) The sample means and standard deviations for the Northeast Region and Northwest Region are, respectively,

xNE=56.20,  sNE=12.61,  xNW=48.73,  sNW=11.18

.

Calculate a 90% confidence interval for

NENW

, the difference in the mean energy consumption of Northeast Region and Northwest Region.

Source of Variation     df       Sum of Squares Mean Square F   Groups        872.01    Error            Total    12100.4

Explanation / Answer

a) H0: All the treatments have same mean

Ha: At least one treatment has different mean

b) Here N=12 and Treatments = 3

The complete ANOVA table is shown below:

c) By using excel function =FDIST(0.758,2,9), the p-value is 0.496.

d) Since p-value is greater than 0.10, fail to reject null hypothesis.

e) The critical value of F at 10% level of significance is 3.006. Since test statistic is less than critical value, fail to reject null hypothesis.

Source of Variation     df       Sum of Squares Mean Square F   Groups 3-1=2 872.01*2=1744.02 872.01 872.01/1150.7=0.758 Error 12-3=9 12100.4-1744.02=10356.38 10356.38/9=1150.7 Total 9+2=11 12100.4

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