Answer to Question #124239 in Statistics and Probability for Evelyn Teye

Question

The fuel consumption of a new model of car manufactured by Kantanka Automobile Limited is being tested.In one trial, 50 cars chosen at random were driven under identical conditions and the distance ,xkm on 1liter of petrol were recorded . The results gave the following totals.
summation of x =525 , summation of x2=5625.
Calculate the 95% confidence interval for the average petrol consumption , in kilometer per liter ,of car of this type.

Answer 1

$\text{The confidence interval for the mean }\\ \text{can be calculated as follows:}\\ \bar x±Z_{\frac{α}{2}}(\frac{s}{\sqrt{n}}),\\ \bar x=\frac{\sum x}{n}=\frac{525}{50}=10.5\\ S^{2}=\frac{1}{n-1}(\sum x^{2} -\frac{(\sum x)^{2}}{n})\\ S^{2}=\frac{1}{49}(5625 -\frac{(525)^{2}}{50})\approx 2.296\\ s=\sqrt{s^2}\approx 1.52,\\ Z_{\frac{α}{2}}=Z_{\frac{0.05}{2}}=Z_{0.025}=1.96,\\ \text{ the lower limit }= 10.5-1.96(\frac{1.52}{\sqrt{50}})\\ ≈10.079,\\ \text{ the upper limit} = 10.5+1.96(\frac{1.52}{\sqrt{50}})\\ ≈10.921,\\ \text{so the confidence interval is}\\ 10.079\leq \bar x \leq 10.921\\$