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Among ​51 to 56-year-olds, ​36% say they have while under the influence of . Suppose ​- to ​-year-olds are selected at random. Complete parts​ (a) through​ (d) below.

User KZiovas
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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.

User Nida Amin
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