A new fuel injection system has been engineered for pickup trucks. The new system and the old system both produce about the same average miles per gallon. However, engineers question which system (old or new) will give better consistency in fuel consumption (miles per gallon) under a variety of driving conditions. A random sample of 41 trucks were fitted with the new fuel injection system and driven under different conditions. For these trucks, the sample variance of gasoline consumption was 52.4. Another random sample of 25 trucks were fitted with the old fuel injection system and driven under a variety of different conditions. For these trucks, the sample variance of gasoline consumption was 34.9. Test the claim that there is a difference in population variance of gasoline consumption for the two injection systems. Use a 5% level of significance. How could your test conclusion relate to the question regarding the consistency of fuel consumption for the two fuel injection systems?
(a) What is the level of significance?
State the null and alternate hypotheses.
H_{o}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} > σ_{2}^{2}H_{o}: σ_{1}^{2} > σ_{2}^{2}; H_{1}: σ_{1}^{2} = σ_{2}^{2} H_{o}: σ_{2}^{2} = σ_{1}^{2}; H_{1}: σ_{2}^{2} > σ_{1}^{2}H_{o}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} ≠ σ_{2}^{2}
(b) Find the value of the sample F statistic. (Round your
answer to two decimal places.)
What are the degrees of freedom?
df_{N} | |
df_{D} |
What assumptions are you making about the original
distribution?
The populations follow independent chi-square distributions. We have random samples from each population. The populations follow independent normal distributions. The populations follow independent normal distributions. We have random samples from each population. The populations follow dependent normal distributions. We have random samples from each population.
(c) Find or estimate the P-value of the sample test
statistic.
P-value > 0.200 0.100 < P-value < 0.200 0.050 < P-value < 0.100 0.020 < P-value < 0.050 0.002 < P-value < 0.020 P-value < 0.002
(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis?
At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the
application.
Fail to reject the null hypothesis, there is sufficient evidence that the variance in consumption of gasoline is greater in the new fuel injection systems. Reject the null hypothesis, there is insufficient evidence that the variance in consumption of gasoline is greater in the new fuel injection systems. Reject the null hypothesis, there is sufficient evidence that the variance in consumption of gasoline is different in both fuel injection systems. Fail to reject the null hypothesis, there is insufficient evidence that the variance in consumption of gasoline is different in both fuel injection systems.
The statistical software output for this problem is:
Hence,
a) Level of significance = 0.05
Hypotheses: H_{o}: σ_{1}^{2}= σ_{2}^{2}; H_{1}: σ_{1}^{2} ≠ σ_{2}^{2}
b) Sample test statistic = 1.50
dfN = 40
dfD = 24
The populations follow independent normal distributions. We have random samples from each population.
c) P-value > 0.200
d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
e) Fail to reject the null hypothesis, there is insufficient evidence that the variance in consumption of gasoline is different in both fuel injection systems.
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