The average speed of motor vehicles on an expressway is being studied. Assume th

Question

The average speed of motor vehicles on an expressway is being studied. Assume that the standard deviation of vehicle speed is known to be 6 km/h. (a) Suppose observations on 60 motor vehicles yielded a sample mean of 100 km/h. Determine two- sided 95% confidence intervals of the mean speed. (Assume a Normal distribution). (b) In part (a), how many additional vehicles' speeds should be observed such that the mean speed can be estimated to within plusminus 1 km/h with 99% confidence? (c) Suppose two observers Jeremy and Jemima are assigned to collect data on the speed of vehicles on this expressway. After each person has separately observed 25 vehicles, what is the probability that Jeremy's sample mean will exceed Jemima's sample mean by 1 plusminus 0.5 km/h? (d) Jeremy has observed 210 vehicles and found a sample mean of 105km/h while Jemima has separately observed 180 vehicles and found a sample mean of 103km/h. Use the 0.01 level of significance to test the null hypothesis that these two population means are equal against the alternative hypothesis that they are not equal.

Explanation / Answer

a) here std error=std deviation/(n)1/2 =0.7746

for 95% CI,z ; t=1.96

hence confidence interval =sample mean -/+ z*std error =98.4818 ; 101.5182

b)here margin of error E =1

hence sample size =(z*std deviation/E)2 =~139

hence 139-60=79 additional speeds needed

c)here sample mean differnce =0

and std error of differnece =(62/25+62/25)1/2 =1.697

hence P(-0.5<X<1.5)= P(-0.5/1.697<Z<1.5/1.697) =P(-0.2946 <Z<0.8839) =0.8116-0.3841=0.4275

d)here std errir of mean =(62/210 +62/180)1/2 =0.6094

hence test stat z=(X1-X2)/std error =(105-103)/0.6094 =3.2817

for above test stat p vlaue=0.0010

as p vlaue is less then 0.01 level we reject null hypothesis and conclude that mean are not equal

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