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

Respuesta :

Answer:

b) 0.2961

c) 0.2251

d) Mean = 11.25, Standard deviation =  1.667  

Step-by-step explanation:

We are given the following information:

We treat trucks undergoing a brake inspection passin as a success.

P( trucks undergoing a brake inspection passes the test) = 75% = 0.75

a) Conditions for binomial probability distribution

  1. There are n independent trial.
  2. Each trial have a success probability p
  3. The probability of success is same for all trials.

Then the number of trucks undergoing a brake inspection follows a binomial distribution, where

[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 15

b) P(proportion of groups will between 8 and 10 trucks pass the inspection)

We have to evaluate:

[tex]P(8\leq x \leq 10) = P(x = 8) + P(x = 9) + P(x = 10)\\= \binom{15}{8}(0.75)^8(1-0.75)^7 + \binom{15}{9}(0.75)^9(1-0.75)^6 + \binom{15}{10}(0.75)^{10}(1-0.75)^5\\= 0.0393 + 0.0917 + 0.1651\\= 0.2961[/tex]

c) P( exactly 3 trucks fail the inspection)

p = 0.25

[tex]P(x = 3)\\= \binom{15}{3}(0.25)^3(1-0.25)^{12}\\=0.2251[/tex]

d) Mean and standard deviation

[tex]\mu = np = 15(0.75) = 11.25\\\sigma = \sqrt{np(1-p)} = \sqrt{15(0.75)(1-0.75)} = 1.667[/tex]