Final answer:
To find the probability that in at most 2 cases, a traffic fatality involved an alcohol-impaired or intoxicated driver, or a non-occupant, we can use the binomial distribution formula. The probability that in at most 2 cases it involved an alcohol-impaired or intoxicated driver, or a non-occupant, is 32.0%. The mean and standard deviation of the variable are 3.2 and approximately 1.472, respectively.
Step-by-step explanation:
To find the probability that in at most 2 cases, a traffic fatality involved an alcohol-impaired or intoxicated driver, or a non-occupant:
A) We can use the binomial distribution formula. Let's first calculate the probability that it involved an alcohol-impaired or intoxicated driver, or a non-occupant, in exactly 0 cases: P(X=0) = (0.40)^0 * (1-0.40)^8 = 0.0%
Then, calculate the probability that it involved an alcohol-impaired or intoxicated driver, or a non-occupant, in exactly 1 case: P(X=1) = 8 * (0.40)^1 * (1-0.40)^7 = 8.96%
Finally, calculate the probability that it involved an alcohol-impaired or intoxicated driver, or a non-occupant, in exactly 2 cases: P(X=2) = 28 * (0.40)^2 * (1-0.40)^6 = 23.04%
The probability that in at most 2 cases it involved an alcohol-impaired or intoxicated driver, or a non-occupant, is the sum of the probabilities from 0 to 2 cases: P(X≤2) = P(X=0) + P(X=1) + P(X=2) = 0.0% + 8.96% + 23.04% = 32.0%
B) The mean of the binomial distribution is given by the formula E(X) = n * p, where n is the number of trials and p is the probability of success. In this case, n = 8 and p = 0.40. Therefore, the mean is E(X) = 8 * 0.40 = 3.2.
The standard deviation of the binomial distribution is given by the formula σ(X) = sqrt(n * p * (1-p)). Plugging in the values, we get σ(X) = sqrt(8 * 0.40 * (1-0.40)) ≈ 1.472.