Okay, let's break this down step-by-step:
(a) Define the random variable X to be the number who say they have driven while under the influence of alcohol.
X = the number who say they have driven under the influence of alcohol
(b) State the distribution of X.
Since a simple random sample of n 51 to 56-year-olds is being selected, X follows a binomial distribution with parameters:
n = the sample size
p = 0.36 (the probability that a 51 to 56-year-old says they have driven under the influence)
(c) Find the probability that at least 2 say they have driven under the influence of alcohol.
Let's define some values:
n = 5 (the sample size given)
p = 0.36
X = the number who say they have driven under the influence
We want to find P(X ≥ 2)
Using the binomial distribution formula:
P(X ≥ 2) = 1 - P(X ≤ 1)
= 1 - (binomcdf(n,p,0) + binomcdf(n,p,1))
= 1 - (0.2034 + 0.3806)
= 1 - 0.584
= 0.416
Therefore, the probability that at least 2 say they have driven under the influence is 0.416.
(d) Find the probability that at most 2 say they have driven under the influence of alcohol.
We want to find P(X ≤ 2)
Using the binomial distribution formula again:
P(X ≤ 2) = binomcdf(n,p,0) + binomcdf(n,p,1) + binomcdf(n,p,2)
= 0.2034 + 0.3806 + 0.2592
= 0.8432
Therefore, the probability that at most 2 say they have driven under the influence is 0.8432.