7.53 Urban planners will use electronic traffic counters to count the number of

Question

7.53

Urban planners will use electronic traffic counters to count the number of vehicles that travel on a specific road. This information can be used to identify roads that need to be improved or expanded based on traffic needs. Suppose that, from historical studies, Duncan Road in Kent County averages, 2,165 vehicles per day with a standard deviation of 485 vehicles per day. A traffic counter was used on this road on 33 days that were randomly slected.

a. What is the probability that the sample mean is less than 2,000 vehicles per day?

b. What is the probability that the sample mean is more than 2,200 vehicles per day?

c. What is the probability that the sample mean is between 2,100 and 2,300 vehicles per day

Explanation / Answer

Using central limit theorem the sampling distribution of sample mean is also normal with mean mu=2165 and standard deviation=sigma/root over n=485/root over 33=84.42

a.For X=2000, z=(x-mu)/sigma=(2000-2165)/84.42=-1.95

P(X<2000)=P(z<-1.95)=0.0256

b.For X=2200, z=(2200-2165)/84.42=0.41

P(X>2200)=P(z>0.41)=1-P(z<0.41)=1-0.6591=0.3409

c.For X=2100, z=(2100-2165)/84.42=-0.76

For X=2300, z=(2300-2165)/84.42=1.59

P(2100<X<2300)=P(X<2300)-P(X<2100)

=P(z<1.59)-P(z<-0.76)

=0.9441-0.2236

=0.7206

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