You are the manager of the Metropolitan Bridge and Tunnel Authority. You want to

Question

You are the manager of the Metropolitan Bridge and Tunnel Authority. You want to see where the traffic that uses Memorial Bridge originates from. Based upon past history, you hypothesize that 40 percent of drivers live within 5 miles of the bridge; 30 percent live within the next 5 miles; 20 percent within the next 5 miles; and the remaining drivers even farther away. You randomly sample 200 bridge users and ask them where they started their trip. They answer the following: 90 within 5 miles; 60 within 10 miles but more than 5 miles; 30 within 15 miles but more than 10 miles; and the remaining drivers even farther away. Test your hypothesis at the 1 percent alpha level.

Explanation / Answer

the whole population is divided into 4 mutually exclusive and exhaustive groups.

1st group consists of driver living within 5 miles of the bridge.

2nd group consists of drivers living within next 5 miles of the bridge

3rd group consists of drivers living within next 5 miles of the bridge

4th group consists of drivers living even further away

let pi be the proportion of drivers living in the ith group.

Based upon past history, you hypothesize that 40 percent of drivers live within 5 miles of the bridge; 30 percent live within the next 5 miles; 20 percent within the next 5 miles; and the remaining drivers even farther away.

so we need to test

H0: p1=0.4 p2=0.3 p3=0.2 p4=0.1 vs H1: not H0

level of significance=alpha=1%=0.01

to test the hypothesis we have a random sample of size n=200 out of which f1=90 lives in group 1

f2=60 lives in group 2 f3=30 lives in group 3 f4=200-90-60-30=20 lives in group 4

the test statistic is given as

T=(f1-n*p1)2/n*p1+(f2-n*p2)2/n*p2+(f3-n*p3)2/n*p3+(f4-n*p4)2/n*p4 which under H0 follows a chi square distribution with df=k-1 since n=200 is quite large. k is the number of groups=4

H0 is rejected iff t>chialpha;k-1   where t is the observed value of T and chialpha;k-1 is the upper alpha point of a chi square distribution with k-e degrees of freedom.

now t=(90-200*0.4)2/(200*0.4)+(60-200*0.3)2/(200*0.3)+(30-200*0.2)2/(200*0.2)+(20-200*0.1)2/(200*0.1)=3.75

chi0.01;3=11.34487

so t<chi0.01;3

hence we fail to reject H0 at 1% level of significance, hence the conclusion is that based on past history the claim that 40 percent of drivers live within 5 miles of the bridge; 30 percent live within the next 5 miles; 20 percent within the next 5 miles; and the remaining drivers even farther away is true

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