Question

A major oil company has developed a new gasoline additive that is supposed to increase mileage. To test this hypothesis, ten cars are randomly selected. The cars are driven both with and without the additive. The results are displayed in the following table. Can it be concluded, from the data, that the gasoline additive does significantly increase mileage? Let d = (gas mileage with additive)–(gas mileage without additive). Use a significance level of a = 0.01 for the test. Assume that the gas mileages are normally distributed for the population of all cars both with and without the additive. Car 3 4 5 6 7 8 9 10 Without additive 23.6 14.9 26 27.1 23.8 14.7 10.3 21.7 11 18.3 With additive 24.3 16.8 27.3 30.2 27.3 16.9 12.2 24.4 12.7 21.5

Answer 1

Let us denote the difference

d = gas mileage with additive - gas mileage without additive

Let Hd denotes the population mean of differnce. Hypotheses : To test null hypothesis Ho Hd = 0 against alternative hypothesis H1 :pid > 0 This is a one sided right tailed test as alternative hypothesis H1 contains only > sign. Let d denotes the sample mean of difference, sa denotes the sample standard deviation of difference and n denotes the sample size. Here, d = 2.2200, så = 0.8040, sd = 0.8967, n= 10 The test statistic can be written as: vnd – 0.0) t= sd which under H, follows a t distribution with n - 1 df. Decision rule / Rejection region : We reject H, at 0.01 level of significance if P-value < 0.01 or if tobs > t0.01,n–1 = 2.821438 0.4 0.3 density 0.2 0.1 - 0.0- Now, Value of test statistic: The value of the test statistic tobs = V10 (2.2200 - 0) 0.8967 7.829336 7.83 associated degrees of freedom = 9 The critical value = critical = 10.01,9 = 2.821438 2.821 P-value = P(ty > 7.829336) = 1- P(tg < 7.829336) = 1-0.999987 = 0.0000132 0.0000 Conclusion : Since p-value < 0.01 and tobs > tcritical = 2.821438, so we reject Ho at 0.01 level of significance Hence, we can conclude that mean difference is significantly greater than 0

There is sufficient evidence to conclude that new gasoline additive significantly increases mileage.