The heating bills for a selected sample of houses using various forms of heating are given below (values are in dollars):

- Gas Heated Homes: 83, 80, 82, 83, 82
- Central Electric: 90, 88, 87, 82, 83
- Heat Pump: 81, 83, 80, 82, 79

Tasks:

a. State the null and alternative hypotheses.

b. Compute the sum of squares between treatments.

c. Compute the mean square between treatments.

d. Compute the sum of squares due to error.

e. Compute the mean square due to error.

f. Set up the ANOVA table for this problem.

g. At [tex]\alpha = 0.05[/tex], test to see if there is a significant difference among the average bills of the homes.

Null Hypothesis (H0): No significant difference among average bills of homes with different heating. Alternative Hypothesis (Ha): Significant difference among average bills.

ANOVA test confirms significant difference in average bills for different heating methods at 0.05 significance.

a. Null Hypothesis (H0): No significant difference among average bills of homes with different heating. Alternative Hypothesis (Ha): Significant difference among average bills.

b. Calculate group means: Gas (82), Electric (86), Heat Pump (81). Calculate overall mean (83).

Sum of Squares Between Treatments (SSB) = 26.

c. Mean Square Between Treatments (MSB) = 13 (SSB / Degrees of Freedom).

d. Sum of Squares Due to Error (SSE) = 38.

e. Mean Square Due to Error (MSE) = 3.167 (SSE / Degrees of Freedom Within).

f. ANOVA Table:

Source | DF | SS | MS | F

Between | 2 | 26 | 13 | MSB / MSE

Within | 12 | 38 | 3.167

Total | 14 | 64

g. Test Statistic (F) = 4.11. Critical value (α = 0.05) = 3.88. Since 4.11 > 3.88, reject H0.

Conclusion: At 0.05 significance, evidence suggests significant difference in average bills for different heating methods.

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