Suppose that 75% of all trucks undergoing a brake inspection at a certain inspection facility pass the inspection. Consider groups of 15 trucks and let X be the number of trucks in a group that have passed the inspection.

a. Verify that this is a binomial distribution
b. For what proportion of groups will between 8 and 10 trucks (inclusive) pass the inspection?
c. For what proportion of groups will exactly 3 trucks fail the inspection?
d. Find the mean and the standard deviation of the random variable X

a)For this case, we assume that we have independence bewteen the vents and each trial have a probability of success , so then the variable X follows a rdon variable

b)  

And adding we got: 0.1698

c)  

d) The expected value is given by:

The deviation i given by:

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".  

Let X the random variable of interest, on this case we now that:  

The probability mass function for the Binomial distribution is given as:  

Where (nCx) means combinatory and it's given by this formula:  

Part a

For this case, we assume that we have independence bewteen the vents and each trial have a probability of success , so then the variable X follows a rdon variable

Part b

And adding we got: 0.1698

Part c

Part d

The expected value is given by:

The deviation i given by:

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.