Answer:
1. Yes, there is sufficient evidence to support the claim that student cars are older than faculty cars.
2. The 98% confidence interval for the difference between the two population means is [1.432 years, 3.968 years].
Step-by-step explanation:
We are given that randomly selected 110 student cars to have ages with a mean of 8 years and a standard deviation of 3.6 years, while randomly selected 75 faculty cars to have ages with a mean of 5.3 years and a standard deviation of 3.7 years.
Let
= <em>mean age of student cars.</em>
= <em>mean age of faculty cars.</em>
So, Null Hypothesis,
:
{means that the student cars are younger than or equal to faculty cars}
Alternate Hypothesis,
:
{means that the student cars are older than faculty cars}
(1) The test statistics that will be used here is <u>Two-sample t-test statistics</u> because we don't know about the population standard deviations;
T.S. =
~
![t_n_1_+_n_2_-_2](https://tex.z-dn.net/?f=t_n_1_%2B_n_2_-_2)
where,
= sample mean age of student cars = 8 years
= sample mean age of faculty cars = 5.3 years
= sample standard deviation of student cars = 3.6 years
= sample standard deviation of student cars = 3.7 years
= sample of student cars = 110
= sample of faculty cars = 75
Also,
=
= 3.641
So, <u><em>the test statistics</em></u> =
~ ![t_1_8_3](https://tex.z-dn.net/?f=t_1_8_3)
= 4.952
The value of t-test statistics is 4.952.
Since the value of our test statistics is more than the critical value of t, so <u><em>we have sufficient evidence to reject our null hypothesis</em></u> as it will fall in the rejection region.
Therefore, we support the claim that student cars are older than faculty cars.
(2) The 98% confidence interval for the difference between the two population means (
) is given by;
98% C.I. for (
) = ![(\bar X_1-\bar X_2) \pm (t_(_\frac{\alpha}{2}_) \times s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} })](https://tex.z-dn.net/?f=%28%5Cbar%20X_1-%5Cbar%20X_2%29%20%5Cpm%20%28t_%28_%5Cfrac%7B%5Calpha%7D%7B2%7D_%29%20%5Ctimes%20s_p%20%5Ctimes%20%20%5Csqrt%7B%5Cfrac%7B1%7D%7Bn_1%7D%2B%5Cfrac%7B1%7D%7Bn_2%7D%20%7D%29)
= ![(8-5.3) \pm (2.326 \times 3.641 \times \sqrt{\frac{1}{110}+\frac{1}{75} })](https://tex.z-dn.net/?f=%288-5.3%29%20%5Cpm%20%282.326%20%5Ctimes%203.641%20%5Ctimes%20%20%5Csqrt%7B%5Cfrac%7B1%7D%7B110%7D%2B%5Cfrac%7B1%7D%7B75%7D%20%7D%29)
= ![[2.7 \pm 1.268]](https://tex.z-dn.net/?f=%5B2.7%20%5Cpm%201.268%5D)
= [1.432, 3.968]
Here, the critical value of t at a 1% level of significance is 2.326.
Hence, the 98% confidence interval for the difference between the two population means is [1.432 years, 3.968 years].