Question

How many people in a car? A study of rush-hour traffic in San Francisco counts the number of people in each car entering a freeway at a suburban interchange. Suppose that this count has mean 1.5 and standard deviation 0.75 in the population of all cars that enter at this interchange during rush hours. (a) Could the exact distribution of the count be Normal? Why or why not? (b) Traffic engineers estimate that the capacity of the interchange is 700 cars per hour. Find the probability that 700 randomly selected cars at this freeway entrance will carry more than 1075 people. Show your work. (Hint: Restate this event in terms of the mean number of people per car.)

Answer 1

Answer:

a) No. It is not normal.

b) The probability that 700 randomly selected cars at this freeway entrance will carry more than 1075 people is 0.104

Step-by-step explanation:

(a) Could the exact distribution of the count be Normal?

The exact distribution of the number of people in each car entering a freeway at a suburban interchange is not normal. Because the count is discrete and can assume values bigger or equal to one.

(b) The probability that 700 randomly selected cars at this freeway entrance will carry more than 1075 people.

The probability we seek is the cars carrying people with mean more than $\frac{1075}{700}=1.5357$

That is P(z>z*) where z* is the z-score of 1.5357.

z* can be calculated using the equation:

z*=$\frac{X-M}{\frac{s}{\sqrt{N} } }$ where

• X is the mean value wee seek for its z-score (1.5357)

• M is the average count of people entering a freeway at a suburban interchange. (1.5)

• s is the standard deviation of the count (0.75)

• N is the sample size (700)

Thus z*=$\frac{1.5357-1.5}{\frac{0.75}{\sqrt{700} } }$ ≈ 1.26

We have P(z>1.26)=1-P(z≤1.26)= 1-0.896 = 0.104