work Chapter 6
Question 48, 6.6.8-T ,
HW Score: $17.65 \%, 9$ of 51 points
Points: 0 of 1
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If $n p \geq 5$ and $n q \geq 5$, estimate $P($ at least 7 ) with $n=13$ and $p=0.5$ 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. $P($ at least 7$)=$
(Round to three decimal places as needed.)
B. The normal distribution cannot be used.

Step 1 ：The problem is asking to estimate the probability of getting at least 7 successes in a binomial distribution with parameters n=13 and p=0.5 using the normal distribution as an approximation. The conditions for using the normal approximation are met since both np and nq are greater than 5.

Step 2 ：To use the normal approximation, we need to calculate the mean and standard deviation of the binomial distribution. The mean of a binomial distribution is np and the standard deviation is \(\sqrt{npq}\).

Step 3 ：Then, we can use the normal distribution to approximate the probability. However, we need to apply the continuity correction since we are approximating a discrete distribution with a continuous one. So, instead of finding P(X >= 7), we find P(X > 6.5).

Step 4 ：Finally, we standardize and use the standard normal distribution to find the probability.

Step 5 ：The calculations are as follows: n = 13, p = 0.5, q = 0.5, mean = 6.5, std_dev = \(\sqrt{1.8027756377319946}\), x = 6.5, z = 0.0, prob = 0.5

Step 6 ：This means that there is a 50% chance of getting at least 7 successes in a binomial distribution with parameters n=13 and p=0.5 when using the normal distribution as an approximation.

Step 7 ：Final Answer: \(P(\text{ at least 7})= \boxed{0.500}\)