iii) Construct a 95% confidence interval for the ratio of two population variances for prices of the two portfolios. Explain if the interval constructed is consistent with the assumption of variances that was stated in (i) above. (6 marks) iv) Using the 95% confidence interval for the mean difference in the output above, can you conclude a difference in prices between the two portfolios? Justify your answer. (2 marks) b) The manufacturer of a fuel additive claims that the use of this additive increased mileage. A random sample of six cars was selected and these cars were driven for one week without the fuel additive and then for one week with the fuel additive. The Minitab output below gives the paired T-Test and CI for these cars without and with the additive. Assume the distance travelled with or without the additive are normally distributed. Paired T-Test and Cl: Without, With Paired T for Without - With N Mean St.Dev SE Mean Without 6 23.40 5.42 2.21 With 6 25.22 5.79 2.36 Difference 6 D 1. 423 0. 581 T-Test of mean difference - 0: T-Value - T Assume that the population distributions are approximately normally distributed, answer the questions below. i) Find the missing values of D and T in the output. (3 marks) i) Is there enough evidence at 5% significance level to conclude whether the use of fuel additive increases the mean mileage of cars? (5 marks)KEY FORMULAS Binomial probability formula P(X = x) =(1-pyx; x=0,12 ....n Poisson probability formula P(X = x)_ex -; x = 0,1, 2, ... x! CONFIDENCE INTERVALS Parameter & description Two-sided (1 - @)100% confidence interval Mean, u, variance, 6 unknown, small samples of =n-1 Difference in means, #1 -#2 , 1+1 variances of =02" and unknown of = 1 + my -2, 5p = (n, - 1)s,+ (12 - 1)s 2 My + 12 - 2 Difference in means, Hy - H2. variances of #62 and unknown di = - (s,2, + 5 2 2 P m -1 12 - 1 Mean difference for paired samples, My of =n-1 where n is no. of pairs Variance, 62 (n -1)s' (n-1)s' of =n-1 Ratio of the variances ,?/62 1 $7 Fa/2: VIV2 V1 = 0 -1, v2 = 02 -1