If np ≥ 5 and nq ≥ 5, estimate P (fewer than 4) with n = 13 and p = 0.5 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.
P (fewer than 4) =
0.0480
To estimate P (fewer than 4) using the normal distribution as an approximation to the binomial distribution, we need to check if np ≥ 5 and nq ≥ 5. In this case, n = 13 and p = 0.5.
First, we calculate np and nq:
np = 13 * 0.5 = 6.5
nq = 13 * (1 - 0.5) = 6.5
Since both np and nq are greater than or equal to 5, we can use the normal distribution as an approximation.
To find P (fewer than 4), we can find the probability of 3 or fewer successes (out of 13) using the normal distribution.
Here's how to calculate it:
1. Calculate the mean (μ) and standard deviation (σ) of the normal distribution using the formula:
μ = np = 6.5
σ = √(npq) = √(6.5 * 0.5 * 0.5) ≈ 1.802
2. Calculate the z-score for the value x = 3 using the formula:
z = (x - μ) / σ = (3 - 6.5) / 1.802 ≈ -1.943
3. Look up the z-score in the standard normal distribution table or use a calculator to find the corresponding cumulative probability.
The cumulative probability for z = -1.943 is approximately 0.026.
4. Since we want the probability of 3 or fewer successes, we need to subtract this probability from 1:
P (fewer than 4) = 1 - 0.026 ≈ 0.974
Therefore, the estimated probability of getting fewer than 4 successes (P fewer than 4) using the normal approximation is approximately 0.974.
Note: If np < 5 or nq < 5, then the normal approximation is not suitable, and you would need to use other methods to estimate the binomial probability.
To determine whether the normal approximation is suitable, we need to check if np ≥ 5 and nq ≥ 5.
Given n = 13 and p = 0.5:
np = 13 * 0.5 = 6.5
nq = 13 * (1 - 0.5) = 6.5
Both np and nq are equal to 6.5, which means the normal approximation is not suitable because np < 5 and nq < 5.
Therefore, we cannot estimate P (fewer than 4) using the normal distribution as an approximation to the binomial distribution.