Question

With gasoline prices increasing, drivers are more concerned with their cars’ gasoline consump- tion. For the past 5 years, a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg^2. Now that his car is 5 years old, he would like to know whether the variability of gas mileage has changed. He recorded the gas mileage from his last eight fill-ups; these are listed here. Conduct a test at a 10% significance level to infer whether the variability has changed.

28 25 29 25 32 36 27 24

Answer 1

The calculated chi -square value =5.7356 Is less than12.01 at 0.1level of significance .

The null hypothesis is accepted.

The past 5 years, a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg^2.

Explanation:

Step:-(1)

Given data 28 25 29 25 32 36 27 24

Sample size 'n' = 8

mean of the sample (x⁻) = ∑x /n =
`(28+ 25+ 29 +25 +32 +36 + 27 + 24)/(8)`

mean of the sample (x⁻) = 28.25

x x- x⁻ (x-x⁻ )²

28 28 - 28.25 = -0.25 0.0625

25 25 -28.25 = -3.25 10.56

29 29-28.25 = 0.75 0.5625

25 25-28.25 = -3.25 10.56

32 32-28.25 = 3.75 14.06

36 36-28.25 = 7.75 60.06

27 27-28.25 = -1.25 1.5625

24 24-28.25 = -4.25 18.06

∑(x-x⁻)² = 115.4875

Step:-(2)

The sample standard deviation

S² = ∑(x-x⁻)²/n-1 =
`(115.4875 )/(7)` = 16.49

By using χ² distribution

χ² =
`(ns^2 )/(variance)`

By above test can be applied only if the population from which sample is drawn normal.

Given data For the past 5 years, a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg^2.

Population variance σ² = 23

Step:-(3)

Null hypothesis :H₀:σ² = 23

Alternative hypothesis :H₁:≠23

`X^(2) = (8 (16.49))/(23)=5.7356`

The calculated chi -square value =5.7356

The degrees of freedom γ=n-1 =8-1=7

The tabulated value of chi -square = 12.01 at 0.1level of significance (check table)

The calculated chi -square value =5.7356 Is less than12.01 at 0.1level of significance .

The null hypothesis is accepted.

Conclusion:-

The null hypothesis is accepted.

The past 5 years, a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg^2.