Question

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

a. Define the parameter of interest.

b. State the relevant hypotheses.

c. Suppose braking distance for the new system is normally distributed with σ = 11. Let ?̅ denote the sample average braking distance for a random sample of 36 observations. Which values of ?̅are more contradictory to H0 than 117.2?

d. What is the P-value in this case? What conclusion is appropriate if α = 0.10?

e. What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the test from part (b) is used?

Answer 1

Given that,
population mean(u)=120
standard deviation, σ =11
sample mean, x =117.2
number (n)=36
null, Ho: μ=120
alternate, H1: μ<120
level of significance, α = 0.1
from standard normal table,left tailed z α/2 =1.282
since our test is left-tailed
reject Ho, if zo < -1.282
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 117.2-120/(11/sqrt(36)
zo = -1.527
| zo | = 1.527
critical value
the value of |z α| at los 10% is 1.282
we got |zo| =1.527 & | z α | = 1.282
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : left tail - ha : ( p < -1.527 ) = 0.063
hence value of p0.1 > 0.063, here we reject Ho
ANSWERS
---------------
a.
the parameter of interest.
population mean(u)=120
standard deviation, σ =11
sample mean, x =117.2
number (n)=36
b.
Z test for single mean
c.
null, Ho: μ=120
alternate, H1: μ<120
test statistic: -1.527
critical value: -1.282
decision: reject Ho
d.
p-value: 0.063
we have enough evidence to support the claim that the true average braking distance at 40 mph under specified conditions is known to be 120 ft.
e.
Given that,
population mean(u)=120
standard deviation, σ =11
sample mean, x =115
number (n)=36
null, Ho: μ=120
alternate, H1: μ!=120
level of significance, α = 0.1
from standard normal table, two tailed z α/2 =1.645
since our test is two-tailed
reject Ho, if zo < -1.645 OR if zo > 1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 115-120/(11/sqrt(36)
zo = -2.727
| zo | = 2.727
critical value
the value of |z α| at los 10% is 1.645
we got |zo| =2.727 & | z α | = 1.645
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : two tailed ( double the one tail ) - ha : ( p != -2.727 ) = 0.006
hence value of p0.1 > 0.006, here we reject Ho
ANSWERS
---------------
null, Ho: μ=120
alternate, H1: μ!=120
test statistic: -2.727
critical value: -1.645 , 1.645
decision: reject Ho
p-value: 0.006
we have enough evidence to support the claim that the true average braking distance is actually 115 ft.