A recent survey in Montana revealed that 70% of the vehicles traveling on highwa

Question

A recent survey in Montana revealed that 70% of the vehicles traveling on highways. where speed limits are posted at 60 miles per hour, were exceeding the limit. Suppose you randomly record the speeds of nine vehicles traveling on US 131 where the speed limit is 60 miles per hour. Let X denote the number of vehicles that were exceeding the limit. a. Find p(X = 9). b. Find p(3 lessthanorequalto X lessthanorequalto 6) c. Find the expected number of vehicles that are traveling on Montana highways and exceeding the speed limit. d. Find the standard deviation of number of vehicles that are traveling on Montana highways and exceeding the speed limit.

Explanation / Answer

We use the binomial distribution if the following requires four assumptions are satisfied:

1) sample size (n) is fixed here n = 9 so this assumption satisfy

2)Each replication of the process results in one of two possible outcomes (success or failure),

here speed above 60 km per hour is success and less 60 km per hour is failure so this assumption also satisfies

3)The probability of success is the same that is constant for each replication, and

Here proportion of speed greater than 60 km per hour is 70% for each vehicle is same and hence constant.

4) The replications(that is trials) are independent,

here that a success in one vehicle does not influence the probability of success in another vehicle.

So all the assumption are valid and we can use Binomial distribution to find the probabilities.

part a)

P(X = 9) = "=BINOMDIST(9,9,0.7,0)" USE THIS EXCEL COMMAND TO FIND BINOMIAL PROBABILITY

So P(X = 9) = 0.040354

b) next we need to find P( 3 <= X <=6) = P(X = 3) + P(X = 4) + P(X =5) + P(X = 6)

=BINOMDIST(3,9,0.7,0)+BINOMDIST(4,9,0.7,0)+BINOMDIST(5,9,0.7,0)+BINOMDIST(6,9,0.7,0)

we get P( 3 <= X <=6) = 0.532858

(c) The expected value that is mean of Binomial distribution is = n * p = 9 * 0.7 = 6.3

d) formula of variance for binomial distribution is = n * p* ( 1 - p) = 9 * 0.7 *( 1 - 0.7) = 1.89

And standard deviation = square root of variance = SQRT(1.89) = 1.3748

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