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

Answer:

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.

Step-by-step explanation:

Step:-(1)

Given data 28 25 29 25 32 36 27 24

Sample size 'n' = 8

mean of the sample (x⁻) = ∑x /n = $\frac{28+ 25+ 29 +25 +32 +36 + 27 + 24}{8}$

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 = $\frac{115.4875 }{7}$  = 16.49

   By using χ² distribution

                           χ²  = $\frac{ns^2 }{variance}$

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

$X^{2} = \frac{8 (16.49)}{23}=5.7356$

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.