Answer:
a) Level of significance α=0.05
Two-tailed test, with null and alternative hypothesis:
![H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu_1-%5Cmu_2%3D0%5C%5C%5C%5CH_a%3A%5Cmu_1-%5Cmu_2%5Cneq%200)
b) Student's t distribution. We assume equal variances for both populations, independent sampled values and populations normally distributed.
Test statistic t=-2.4
c) P-value = 0.018
d) Rejection of the null hypothesis.
The data is statistically significant.
e) There is evidence to conclude there is significant difference in average off-schedule times between the bus lines. The difference we see in the samples seems not due to pure chance.
Step-by-step explanation:
This is a hypothesis test for the difference between populations means.
The claim is that there is a significant difference in average off-schedule times for this bus lines.
Then, the null and alternative hypothesis are:
![H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu_1-%5Cmu_2%3D0%5C%5C%5C%5CH_a%3A%5Cmu_1-%5Cmu_2%5Cneq%200)
The significance level is 0.05.
The sample 1 (bus line A), of size n1=51 has a mean of 53 and a standard deviation of 17.
The sample 2 (bus line B), of size n2=60 has a mean of 60 and a standard deviation of 13.
The difference between sample means is Md=-7.
The estimated standard error of the difference between means is computed using the formula:
Then, we can calculate the t-statistic as:
![t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{-7-0}{2.913}=\dfrac{-7}{2.913}=-2.4](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7BM_d-%28%5Cmu_1-%5Cmu_2%29%7D%7Bs_%7BM_d%7D%7D%3D%5Cdfrac%7B-7-0%7D%7B2.913%7D%3D%5Cdfrac%7B-7%7D%7B2.913%7D%3D-2.4)
The degrees of freedom for this test are:
This test is a two-tailed test, with 109 degrees of freedom and t=-2.4, so the P-value for this test is calculated as (using a t-table):
As the P-value (0.018) 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 there is a significant difference in average off-schedule times for this bus lines.