If np is greater than or equal to 5, and nq is greater than or equal to 5, estimate P(fewer than 8) 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.
PLEASE. I really need help with the steps to solve this. This is my last class and then I graduate. My husband passed away a few weeks ago and I just can't think past my nose. It has taken me four days to do 9 out of my 10 pre-test
I'm sorry to hear about your loss. I understand that this might be a difficult time for you. I'm here to help you with the steps to solve the problem.
To estimate P(fewer than 8), we can use the normal distribution as an approximation to the binomial distribution. However, before we do that, we need to check if the conditions for using the normal approximation are met.
The conditions for using the normal approximation to the binomial distribution are:
1. np >= 5
2. nq >= 5
Let's calculate np and nq first:
np = n * p = 14 * 0.6 = 8.4
nq = n * (1 - p) = 14 * (1 - 0.6) = 5.6
Now, let's check the conditions:
1. np >= 5: Here, np is 8.4 which is greater than 5.
2. nq >= 5: Here, nq is 5.6 which is also greater than 5.
Since both conditions are satisfied (np >= 5 and nq >= 5), we can proceed with using the normal approximation to estimate P(fewer than 8).
To use the normal distribution, we need to calculate the mean (μ) and the standard deviation (σ) of the binomial distribution:
μ = np = 8.4
σ = √(npq) = √(8.4 * (1 - 0.6)) = √3.36 ≈ 1.83
Now, we can use the normal distribution to estimate P(fewer than 8):
P(X < 8) = P(Z < (8 - μ) / σ)
= P(Z < (8 - 8.4) / 1.83)
= P(Z < -0.22)
≈ 0.4129 (using a standard normal distribution table or a calculator)
So, the estimated probability of getting fewer than 8 is approximately 0.4129.
Once again, I'm sorry for your loss, and I hope this explanation helps you in completing your pre-test.