If $$ n p \geq 5 $$ and $$ n q \geq 5 $$, estimate $$ P $$ (more than 5 ) with $$ n=14 $$ and $$ p=0.3 $$ by using the normal distribution as an approximation to the binomial distribution; if $$ n p

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If $$ n p \geq 5 $$ and $$ n q \geq 5 $$, estimate $$ P $$ (more than 5 ) with $$ n=14 $$ and $$ p=0.3 $$ by using the normal distribution as an approximation to the binomial distribution; if $$ n p<5 $$ or $$ n q<5 $$, 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. $$ P $$ (more than 5 ) $$ =\square $$ (Round to four decimal places as needed.) B. The normal distribution cannot be used.

UpStudy AI · Accepted Answer

We first check if the conditions for using the normal approximation to the binomial distribution are met. The conditions are:

- $$ np \geq 5 $$
- $$ nq \geq 5 $$ where $$ q = 1-p $$.

Given:
- $$ n = 14 $$,
- $$ p = 0.3 $$ so that $$ q = 0.7 $$.

Step 1. Compute $$ np $$:
$$
np = 14 \times 0.3 = 4.2
$$
Since $$ 4.2 < 5 $$, the condition $$ np \geq 5 $$ is not satisfied.

Step 2. Although we can compute $$ nq $$:
$$
nq = 14 \times 0.7 = 9.8
$$
this condition is met; however, both conditions must be satisfied to use the normal approximation.

Conclusion: Because $$ np < 5 $$, the normal approximation is not suitable.

Answer: B. The normal distribution cannot be used.
The normal distribution cannot be used.

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