if np >5 and nq>5 estimate P (fewer than 4) with n=13 p=0.4 by using normal distribution as an approximate to the binomial distribution if np <5 or nq< then state that the normal approximation is not suitable
To estimate the probability, P (fewer than 4), we need to check if the normal approximation is suitable to use.
In this case, we have n = 13 and p = 0.4. To determine if the normal approximation is suitable, we need to check if np > 5 and nq > 5, where q = 1 - p.
First, let's calculate np and nq:
np = 13 * 0.4 = 5.2
nq = 13 * (1 - 0.4) = 7.8
Since np > 5 and nq > 5, we can proceed with the normal approximation to estimate the probability.
To use the normal approximation, we need to find the mean (μ) and standard deviation (σ) of the corresponding normal distribution. For a binomial distribution, the mean is μ = np and the standard deviation is σ = √(npq).
μ = 13 * 0.4 = 5.2 (mean)
σ = √(13 * 0.4 * (1 - 0.4)) = √(5.2 * 0.6) = √3.12 ≈ 1.7678 (standard deviation)
Now that we have the parameters for the normal distribution, we can approximate the probability using tables or a calculator.
To estimate P (fewer than 4), we need to calculate the cumulative probability up to 3, which can be denoted as P(X < 4). We can rewrite it as P(X ≤ 3).
Using the normal distribution approximation, we convert the binomial distribution to a standard normal distribution by standardizing the values. We use the z-score formula: z = (x - μ) / σ, where x is the value we want to find the probability for.
For P(X ≤ 3), we can calculate the z-score as follows:
z = (3 - μ) / σ
Substituting the values:
z = (3 - 5.2) / 1.7678 ≈ -1.243 (approximately)
Now, using a standard normal distribution table or a calculator, find the cumulative probability for z = -1.243. The result will give you the estimated probability.
Note:
If np < 5 or nq < 5, the normal approximation is not suitable, and you would need to use the binomial distribution directly. However, in this case, since np > 5 and nq > 5, we can use the normal approximation.