14. Safety Engineering. Airbag Deaths, and the Dilemma of the Two-Tail Test In t

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14. Safety Engineering. Airbag Deaths, and the Dilemma of the Two-Tail Test In the late 1990's, Chrysler was being investigated by the U.S. Department of Transportation (USDT) for too high of an incidence in children's deaths due to accidental passenger-side airbag deployments in mini-vans (such as a very slow "fender-bender" that would accidentally trigger the airbag). This was during an era when passenger-side airbags did not have the weight detection technology in the front passenger seat just yet. It was known at the time, that of the 1.5 million minivans on the road with passenger-side airbags, approximately 65% were Chrysler products (Dodge Caravan, Plymouth Voyager, and Chrysler Town & Country) Of the nationwide 20 children's deaths due to accidental passenger-side airbag deployment, 17 of those deaths were in Chrysler products. The question asked: Was this percentage higher than the market share of 65% or simply higher just due to random variability? In other words, was Chrysler at fault here because of defective passenger-side airbags? (Note that a conservative guideline for minimum sample size for a z proportional test would suggest it would have been invalid since there were less than 10 "failures", that is, (1-65)*20 7. However, the more liberal guideline of 5 or more failures/successes" would accommodate this requirement.) To make things more interesting, suppose you are an attorney representing the families who had lost a loved one in an accidental airbag deployment. In a potential lawsuit against Chrysler, you need to check out the pros and cons of running the hypothesis test as a right-tail test rather than a two-tail test. Calculate the P-values for both types of hypothesis tests using the level of significance -0.05. Show your two different results and then state which test you would prefer to run as the attorney and why Two-tail test P-value: Right-tail test P-value: Decision of your preference and why

Explanation / Answer

14.

Given that,
possibile chances (x)=17
sample size(n)=20
success rate ( p )= x/n = 0.85
success probability,( po )=0.65
failure probability,( qo) = 0.35
null, Ho:p=0.65  
alternate, H1: p!=0.65
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.85-0.65/(sqrt(0.2275)/20)
zo =1.8752
| zo | =1.8752
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =1.875 & | z | =1.96
make decision
hence value of |zo | < | z | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 1.87523 ) = 0.06076
hence value of p0.05 < 0.0608,here we do not reject Ho
ANSWERS
---------------
null, Ho:p=0.65
alternate, H1: p!=0.65
test statistic: 1.8752
critical value: -1.96 , 1.96
decision: do not reject Ho
p-value: 0.06076

Given that,
possibile chances (x)=17
sample size(n)=20
success rate ( p )= x/n = 0.85
success probability,( po )=0.65
failure probability,( qo) = 0.35
null, Ho:p=0.65  
alternate, H1: p>0.65
level of significance, = 0.05
from standard normal table,right tailed z /2 =1.64
since our test is right-tailed
reject Ho, if zo > 1.64
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.85-0.65/(sqrt(0.2275)/20)
zo =1.8752
| zo | =1.8752
critical value
the value of |z | at los 0.05% is 1.64
we got |zo| =1.875 & | z | =1.64
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: right tail - Ha : ( p > 1.87523 ) = 0.03038
hence value of p0.05 > 0.03038,here we reject Ho
ANSWERS
---------------
null, Ho:p=0.65
alternate, H1: p>0.65
test statistic: 1.8752
critical value: 1.64
decision: reject Ho
p-value: 0.03038

we conclude that both types of tests are right tailed and two tailed are significant at 0.05 level.
we need prefer pros and cons of hypothesis test of right tailed rather than two tailed because right tailed is reject the null hypothesis and two tailed is do not reject the null hypothesis

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