Answer to Question #164566 in Statistics and Probability for Dean

Question

A transportation company plans to buy tires in volume. They are choosing between two brands of tires, A and B. In order to come up with a wise decision, a survey was conducted between these two brands of tires.

$$\begin{matrix} & \text{BRAND A} & \text{BRAND B}\\ \text{average mileage} & 40,000\ km & 38,000\ km\\ \text{standard deviation} & 4,500\ km & 3,900\ km \end{matrix} $$

Is Brand A better than Brand B; using a 0.05 level?

Answer 1

Let us test the null hypothesis $H_0:\mu_1=\mu_2$ against the alternative hypothesis $H_1:\mu_1>\mu_2$ .

We use the statistic $Z=\frac{\bar{X}_A-\bar{X}_B}{\sqrt{\sigma_x^2+\sigma_y^2}}$. If $X_A$ and $X_B$ are normally distributed, then Z has (approximately) normalized Gauss distribution, and the Laplace function $\Phi(x)$ is its CDF (cumulative distribution function). The critical domain corresponding to a statistical significance $\alpha=0.05$ is $\Phi(Z)>1-\alpha=0.95$ or $Z>\Phi^{-1}(0.95)=1.64$

$Z=\frac{40-38}{\sqrt{4.5^2+3.9^2}}=0.336<1.64$

Therefore, the hypothesis that $\mu_1=\mu_2$ can not be rejected at the significance level 0.05.