Question

A researcher compares two compounds (1 and 2) used in the manufacture of car tires that are designed to reduce braking distances for SUVs equipped with the tires. SUVs equipped with tires using compound 1 have a mean braking distance of 62 feet and a standard deviation of 10.6 feet. SUVs equipped with tires using compound 2 have a mean braking distance of 68 feet and a standard deviation of 13.9 feet. Suppose that a sample of 77 braking tests are performed for each compound. Using these results, test 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. Let μ1 be the true mean braking distance corresponding to compound 1 and μ2 be the true mean braking distance corresponding to compound 2. Use the 0.05 level of significance.

Step 1 of 4: State the null and alternative hypotheses for the test.

Step 2 of 4: Compute the value of the test statistic. Round your answer to two decimal places.

Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0H0. Round the numerical portion of your answer to three decimal places.

Step 4 of 4: Make the decision for the hypothesis test.

Answer 1

Given that,
mean(x)=62
standard deviation , s.d1=10.6
number(n1)=77
y(mean)=68
standard deviation, s.d2 =13.9
number(n2)=77
null, Ho: u1 = u2
alternate, H1: u1 < u2
level of significance, α = 0.05
from standard normal table,left tailed t α/2 =1.665
since our test is left-tailed
reject Ho, if to < -1.665
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =62-68/sqrt((112.36/77)+(193.21/77))
to =-3.012
| to | =3.012
critical value
the value of |t α| with min (n1-1, n2-1) i.e 76 d.f is 1.665
we got |to| = 3.0119 & | t α | = 1.665
make decision
hence value of | to | > | t α| and here we reject Ho
p-value:left tail - Ha : ( p < -3.0119 ) = 0.00176
hence value of p0.05 > 0.00176,here we reject Ho
ANSWERS
---------------
null, Ho: u1 = u2
alternate, H1: u1 < u2
test statistic: -3.012
critical value: -1.665
decision: reject Ho
p-value: 0.00176
we have 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.