A car manufacturer wants to test a new engine to determine whether it meets new

Question

A car manufacturer wants to test a new engine to determine whether it meets new air pollution standard (true mean emission must be less than 20 parts per million of carbon). Thirty-six engines manufactured for testing purposes yield the following summary results of emission levels:

Sample mean =18.50,      Sample std dev = 3

Assume that the normal distribution assumption is satisfied for the sample data.

Using =0.01, conduct a test of   (i) H0: =20; Ha: = 19;      and (ii) H0: =20; Ha: = 18;     (iii) H0: =20; Ha: = 17;  

a.Find the p-value of your test. Does it change as the value of changes in (i), (ii), and (iii).

b. Find power of the test in each of (i), (ii), and (iii). Which test has the highest power?

Explanation / Answer

We have given that,

n=36, xbar=18.50, sd=3, alpha=0.01.

a)

i)testing hypothesis,

H0:=20 vs H1:u=19

Test statistic,

Z=(xbar-)/(sd/n) follow normal distribution (0 ,1)

=(18.50-20)/(3/36)

=-3

P value=P(z<¦-3¦)

=0.0014. (Z table value).

ii) hypothesis, H0: =20; Ha: = 18;

Test statistic,

Z=(xbar-)/(sd/n) follow normal distribution (0 ,1)

=(18.50-20)/(3/36)

=-3

P value=P(z<¦-3¦)

=0.0014. (Z table value)

(iii) H0: =20; Ha: = 17;

Test statistic,

Z=(xbar-)/(sd/n) follow normal distribution (0 ,1)

=(18.50-20)/(3/36)

=-3

P value=P(z<¦-3¦)

=0.0014. (Z table value)

The change the values of then , don't change p value.

b)

Power of test ,1-beta=P(type 2 error)

i) z= (18.50-19)/(3/36)

=-1.

1-beta=P(z>¦-1¦)

= 0.16. Z table value.

ii) z=(18.50-18)/(3/36)

=1

1-beta=0.16.

iii) z= (18.50-17)/(.5)

=3

1- beta= P(z>3)

=0.0014.

Test i and ii has same power and highest power.

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