Question

if np ≥ 5 and nq ≥ 5, estimate p(fewer than 7) with n = 14 and p = 0.6 by using the normal distribution as an approximation to the binomial distribution, if np < 5 or nq < 5, then state that the normal approximation is not suitable.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○ a. p(fewer than 7) =
(round to four decimal places as needed.)
○ b. the normal approximation is not suitable.

Explanation:

Step1: Calculate np and nq

Given \(n = 14\), \(p=0.6\), then \(q = 1 - p=1 - 0.6 = 0.4\).
\(np=14\times0.6 = 8.4\) and \(nq=14\times0.4 = 5.6\). Since \(np\geq5\) and \(nq\geq5\), the normal - approximation to the binomial is suitable.

Step2: Find the mean and standard deviation of the normal - approximation

The mean of the normal - approximation to the binomial is \(\mu=np = 14\times0.6=8.4\).
The standard deviation is \(\sigma=\sqrt{npq}=\sqrt{14\times0.6\times0.4}=\sqrt{3.36}\approx1.8330\).

Step3: Apply the continuity correction

To find \(P(X < 7)\) for the binomial, for the normal approximation we find \(P(X < 6.5)\) (continuity correction).
We calculate the z - score: \(z=\frac{x-\mu}{\sigma}\), where \(x = 6.5\), \(\mu = 8.4\) and \(\sigma\approx1.8330\).
\(z=\frac{6.5 - 8.4}{1.8330}=\frac{-1.9}{1.8330}\approx - 1.04\).

Step4: Find the probability using the standard normal table

We want to find \(P(Z < - 1.04)\).
Looking up the value in the standard - normal table, \(P(Z < - 1.04)=0.1492\).

Answer:

A. \(P(\text{fewer than }7)=0.1492\)