Answer to Question #241087 in Statistics and Probability for Darv

Question

A company is trying to decide which of two types of tires to buy for their trucks. They would like to adopt to brand C unless there is some evidence that brand D is better. An experiment was conducted where 16 tires from each brand were used. The tires were run under similar conditions until they wore out. The results are:

 

Brand C:  x̄₁ = 40,000 kms., s₁ = 5,400 kms.

Brand D:  x̄₂  =38,000 kms., s₂ =3,200 kms.

What conclusions can be drawn?

Answer 1

$n_1=n_2=16 \\ \bar{x_1}=40000 \\ s_1 = 5400 \\ \bar{x_2}= 38000 \\ s_2=3200 \\ H_0: \mu_1 < \mu_2 \\ H_1: \mu_ 1> \mu_2$

When n₁ and n₂ are small and $\sigma_1$ and $\sigma_2$ are unknown, we have to use the two-sample t test for independent random samples from two normal populations having the same unknown variance.

Test-statistic:

$t= \frac{\bar{x_1} - \bar{x_2}}{s_p \sqrt{(1/n_1) + (1/n_2)}} \\ s^2_p= \frac{(n_1-1)s^2_1 +(n_2-1)s^2_2}{n_1+n_2-2} \\ s^2_p= \frac{(16-1)(5400)^2 +(16-1)(3200)^2}{16+16-2} \\ = \frac{4.374 \times 10^8 + 1.536 \times 10^8}{30} \\ = 0.197 \times 10^8 \\ s_p=0.4438 \times 10^4 \\ = 4438.4 \\ t= \frac{40000 -38000}{4438.4 \sqrt{(1/16) + (1/16)}} \\ = \frac{2000}{1569.2} \\ = 1.274$

Reject H₀ if $t≥t_{crit}$

Let use α=0.05

For one-tail test and degree of freedom $df = n_1+n_2-2= 30 \; t_{crit} = 1.697$

$t=1.274 < t_{crit}= 1.697$

Accept H₀ at 0.05 significance level.

Therefore, there is enough evidence to claim that the population mean μ₁ is less than the population mean μ₂. Brand D is better than C.