Question

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 31 trucks were fitted with the new fuel injection system and driven under different conditions. For these trucks, the sample variance of gasoline consumption was 52.7. Another random sample of 24 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 38.5. 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: σ₁² = σ₂²; H₁: σ₁² > σ₂²H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)


What are the degrees of freedom?

dfN
dfD

What assumptions are you making about the original distribution?

The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent chi-square distributions. We have random samples from each population.    The populations follow independent normal distributions. We have random samples from each population.The populations follow independent normal distributions.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.2000.100 < P-value < 0.200    0.050 < P-value < 0.1000.020 < P-value < 0.0500.002 < P-value < 0.020P-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.

Answer 1

Answer:

a) Level of significance:The level of significance is defined as the probability of rejecting a null hypothesis by the test when it is really true, which is denoted as α. That is, P (Type I error) = α.  Here in this problem = 0.05 .

As we want test that whether there is difference in two injection system or not. We set hypothesis as

H0: 12=22 ; H1:12 22

b) Value of sample F statistics is 1.368831

dfN=30 and dfD=23

Assumption : As we are doing F test our assumption about original distribution is The populations follow independent chi-square distributions. We have random samples from each population.

c) we can find p- value by using excel. Using function FDIST and putting values of degrees of freedom we find p - value= 0.220908

Hence P-value>0.200.

d) As p- value is greater than level of significance (p- value>0.05) we can not reject H0. Hence, At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e) As we fail to reject null hypothesis then we can say that variance is same.

i.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. ( All other option are wrong because they reject H0).