Answer:
The null hypothesis is rejected (P-value=0.0
28).
There is enough evidence to support the claim that that Company A tires outlast the tires of Company B by more than 10,000 miles.
Step-by-step explanation:
This is a hypothesis test for the difference between populations means.
The claim is that that Company A tires outlast the tires of Company B by more than 10,000 miles.
Then, the null and alternative hypothesis are:
![H_0: \mu_1-\mu_2=10000\\\\H_a:\mu_1-\mu_2> 10000](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu_1-%5Cmu_2%3D10000%5C%5C%5C%5CH_a%3A%5Cmu_1-%5Cmu_2%3E%2010000)
being μ1: average for Company A and μ2: average for Company B.
The significance level is 0.05.
The sample 1, of size n1=16 has a mean of 63,500 and a standard deviation of 4,000.
The sample 1, of size n1=12 has a mean of 49,500 and a standard deviation of 6,000.
The difference between sample means is Md=14,000.
![M_d=M_1-M_2=63500-49500=14000](https://tex.z-dn.net/?f=M_d%3DM_1-M_2%3D63500-49500%3D14000)
The estimated standard error of the difference between means is computed using the formula:
![s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{4000^2}{16}+\dfrac{6000^2}{12}}\\\\\\s_{M_d}=\sqrt{1000000+3000000}=\sqrt{4000000}=2000](https://tex.z-dn.net/?f=s_%7BM_d%7D%3D%5Csqrt%7B%5Cdfrac%7B%5Csigma_1%5E2%7D%7Bn_1%7D%2B%5Cdfrac%7B%5Csigma_2%5E2%7D%7Bn_2%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B4000%5E2%7D%7B16%7D%2B%5Cdfrac%7B6000%5E2%7D%7B12%7D%7D%5C%5C%5C%5C%5C%5Cs_%7BM_d%7D%3D%5Csqrt%7B1000000%2B3000000%7D%3D%5Csqrt%7B4000000%7D%3D2000)
Then, we can calculate the t-statistic as:
The degrees of freedom for this test are:
This test is a right-tailed test, with 26 degrees of freedom and t=2, so the P-value for this test is calculated as (using a t-table):
![P-value=P(t>2)=0.028](https://tex.z-dn.net/?f=P-value%3DP%28t%3E2%29%3D0.028)
As the P-value (0.028) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that that Company A tires outlast the tires of Company B by more than 10,000 miles.