At fault The car insurance company in Exercise 8 believes that about 0.5% of drivers have an at-fault accidentduring a given year. Suppose the company insures 1355 drivers in that city. a) What are the mean and standard deviation of the number who may have at-fault accidents? b) Can you describe the distribution of these accidents with a Normal model? Explain.

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At fault The car insurance company in Exercise 8 believes that about 0.5% of drivers have an at-fault accidentduring a given year. Suppose the company insures 1355 drivers in that city.  a) What are the mean and standard deviation of the number who may have at-fault accidents?  b) Can you describe the distribution of these accidents with a Normal model? Explain.

Jaylen Fountain · Accepted Answer

Step 1 a.It is given that in exercise 8, there is 0.5% chance that the drivers have an at-fault accident during a given year. In the city, the company insures 1,355 drivers. Properties of Bernoulli trials: In each trial there will be only two possible outcomes, success and failure. The probability of success for each trial is same. Each trial is independent to each other. 10% condition for Bernoulli trials: If the sample size is less than 10% of the population size, then is can be said that the Bernoulli trials must be independent. Binomial probability for Bernoulli trials: The number of successes in a specified number of Bernoulli trials is described by Binomial model. If the number of trials is n, with the probability of success is p and the probability of failure is q=1−p, and X be the event of the number of success in n independent trials, then the binomial probability model is defined as, P(X=x)={(nx)pxqn−x,x=0,1,⋯,n;0&amp;lt;p&amp;lt;1;q=1−p0,otherwise Step 2 With mean is μ=np and standard deviation is σ=npq. It is denoted by X∼Binom(n,p). Let X be the number of at-fault accidents. Checking the Bernoulli properties: In this problem there are only two possible outcome that accident and no accident. Hence 1st condition is satisfied. The probability that the drivers have an at-fault accident during a given year is p=0.005. Hence, 2nd condition is also satisfied. It can be assumed that 1,355 drivers is less than 10% of the whole population of drivers. Hence, using 10% condition rule, the independence condition is also satisfied. Hence, the condition of drivers can be considered as Bernoulli trials. Hence, the distribution of number of at-fault accidents follows Binom(1,355,0.005). Mean: μ=np =1,355×0.005 =6.77 Standard deviation: σ=npq =1,355×0.005×(1−0.005) =6.7411 =2.6 Hence, the mean and the standard deviation of the number who may have at-fault accidents are 6.775 and 2.6. Step 3 b. Normal approximation from Binomial distribution: The success and failure condition: If np≥10 and nq≥10, then the binomial distribution can be approximated as Normal distribution with mean μ=np and variance σ2=npq that is, N(np,npq). Checking success and failure condition: Here, np=1,355×0.005=6.775&amp;lt;10 and nq=1,355(1−0.005)=1,348.225&amp;gt;10 Hence, the success and failure condition is not satisfied. Hence, the distribution of number of accidents cannot be explained by Normal model.

At fault The car insurance company in Exercise 8 believes that about 0.5% of dri

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