Question

determine whether you can use a normal distribution to approximate the binomial distribution. if you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. if you cannot, explain why and use a binomial distribution to find the indicated probabilities. identify any unusual events. explain.
in a survey of u.s. adults, 19% said that they have participated in a clinical trial. you randomly select 190 u.s. adults. find the probability that the number who have participated in a clinical trial is (a) no fewer than 30, (b) more than 38, and (c) between 33 and 43 inclusive.
determine whether a normal distribution can be used to approximate the binomial distribution. choose the correct answer below.
a. no, because np < 5.
b. yes, because both np ≥ 5 and nq ≥ 5.
c. no, because nq < 5.

Explanation:

Step1: Identify values of n and p

Given \(n = 190\) (number of trials) and \(p=0.19\), then \(q = 1 - p=1 - 0.19 = 0.81\).

Step2: Calculate np and nq

\(np=190\times0.19 = 36.1\) and \(nq=190\times0.81 = 153.9\).

Step3: Check the normal - approximation condition

Since \(np = 36.1\geq5\) and \(nq = 153.9\geq5\), a normal distribution can be used to approximate the binomial distribution.

Answer:

B. Yes, because both \(np\geq5\) and \(nq\geq5\)