According to a recent study by fox news, “more than half of U.S. drivers killed

Question

According to a recent study by fox news, “more than half of U.S. drivers killed in car accidents had alcohol or drugs in their system at the time of the crash”. Mr Jenkins decided to test this data for Spokane County car accidents from 2009-2014. During this time period there 53 car accidents in Spokane County involving fatalities. OF these 53 accidents 19 involved a driver who was legally intoxicated. Test the claim at the 5% significance level that less than 50% of the fatal car accidents in Spokane County involve intoxicated drivers. Use the critical value approach

Explanation / Answer

let f denotes the number of accidents involved intoxicated drivers out of the 53 car accidents.

so f~Bin(53,p) where p denotes the probability of a drug involved car accident.

According to a recent study by fox news, “more than half of U.S. drivers killed in car accidents had alcohol or drugs in their system at the time of the crash”.

so H0: p=0.5 vs H1: p>0.5

now E[f/53]=53*p/53=p

V[f/53]=53*p*(1-p)/532=p(1-p)/53

so f/53 is unbiased estimator of p. hence it can be used as to form the test statistic.

now since sample size=53 is very large, we can use CLT and say that

(f/53-p)/sqrt[p(1-p)/53] ~N(0,1)

hence the test statistic is T=(f/53-0.5)/sqrt[0.5*(1-0.5)/53] which under H0 follows N(0,1)

since the alternative hypothesis is greater than type, hence H0 is rejected at 5% level of significance iff

t>z0.05 where t is the observed value of T and z0.05 is the upper 0.05 point of a N(0,1) distribution.

now from sample f=19

so t=-2.0604 and z0.05=1.62

hence t<z0.05

hence H0 is not rejected and the claim is tested to be wrong

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