Although cities encourage carpooling to reduce traffic congestion, most vehicles

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 108 vehicles at random, what is the probability that more than half (that is, 55 or more) carry just one person? (Use the normal approximation.)

Explanation / Answer

A)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.62      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.317686979
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.682313021 [ANSWER]

******************

b)

We first get the z score for the critical value:          
          
x = critical value =    54.5      
u = mean = np =    66.96      
          
s = standard deviation = sqrt(np(1-p)) =    5.044283894      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -2.47012267      
          
Thus, the left tailed area is          
          
P(z >   -2.47012267   ) =    0.993246664 [ANSWER]

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.