Respuesta :
Answer:
a) The null and alternative hypothesis are:
[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0[/tex]
b) t=-1.95
c) Decision rule: Reject H0 if t<-1.654
d) The null hypothesis is rejected.
There is enough evidence to support the claim that the braking distance for SUVs equipped with tires using compound 1 is shorter than the braking distance when compound 2 is used.
Step-by-step explanation:
This is a hypothesis test for the difference between populations means.
The claim is that the braking distance for SUVs equipped with tires using compound 1 is shorter than the braking distance when compound 2 is used.
Then, the null and alternative hypothesis are:
[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0[/tex]
The significance level is 0.05.
The sample 1, of size n1=80 has a mean of 56 and a standard deviation of 6.4.
The sample 2, of size n2=80 has a mean of 58 and a standard deviation of 6.6.
The difference between sample means is Md=-2.
[tex]M_d=M_1-M_2=56-58=-2[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{6.4^2}{80}+\dfrac{6.6^2}{80}}\\\\\\s_{M_d}=\sqrt{0.512+0.545}=\sqrt{1.057}=1.028[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{-2-0}{1.028}=\dfrac{-2}{1.028}=-1.95[/tex]
The degrees of freedom for this test are:
[tex]df=n_1+n_2-1=80+80-2=158[/tex]
Using the critical value approach, the critical value of t for a level of significance of 0.05 and 158 degrees of freedom is tc=-1.654. The rejection region is defined by t<-1.654.
As the test statistic (-1.95) is smaller than the critical value (-1.654), it lies within the rejection region and the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the braking distance for SUVs equipped with tires using compound 1 is shorter than the braking distance when compound 2 is used.
