Question

1. In a certain area of New York State with a large population, 75% of drivers use chains on the car tires for winter driving. A random sample of 16 drivers is taken. You are interested in the number of drivers in the sample that use chains while driving.

   1. Find the probability that at least 11 drivers in the sample use chains.

   2. Find the probability that at most 11 drivers in the sample use chains.

   3. Find the probability that less than 11 drivers in the sample use chains.

   4. Find the mean and standard deviation of the number of drivers in the sample that

      use chains.

   5. Find the probability that the number of drivers in the sample that use chains is

      within 1 standard deviation of its mean.

Answer 1

Given that

Probability that a driver uses chain =P=0.75

Sample size=n=16

Let X is number of drivers out of 16 who uses chain so

X~Binomial (n=16,P=0.75)

So

1)

We gave to find P(X>11)

So

2)

We have to find P(X<11)

So

3)

We have to find P(X<11)

So

4)

Mean =E(X)=n*p=16*0.75=12

5)

We have to find

P(Mean-SD<X<mean+SD)

So here mean -SD =12-1.732=10.268

Mean +SD=12+1.732=13.732

So we have to find

P(10.268<X<13.732)

Now