Question

Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 62% of vehicles on the roads are occupied by just the driver. (Round your answers to four decimal places.)

(a) If you choose 10 vehicles at random, what is the probability that more than half (that is, 6 or more) carry just one person?

(b) If you choose 110 vehicles at random, what is the probability that more than half (that is, 56 or more) carry just one person? (Use the normal approximation.)

Answer 1

Let X be the number of vehicles carry just one person

X ~Binomial( 10, 0.62)

a) More than half means more than 5

b) Whne n = 110

Since the sample size we large enough we will test for normal approximation for binomial distribution

The Conditions are np> 10 and nq > 10

np = 110*0.62 =68.2 >10

nq =110*0.38 =41.8>10

Therefore, this is a good approximation for the binomial distribution.

Mean of Normal distribution is

and the standard deviation is

Now , the probability that more than half (that is, 56 or more) carry just one person is

= 0.4952+0.5 =0.9952