find the normal approximation for the binomial probability that x=5 where n=14 and p=0.3 compare this to the value of p(x=5) found in table 2
To find the normal approximation for the binomial probability, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution, and then convert the binomial probability to a standard normal probability using a z-score.
Step 1: Calculate the mean (μ) of the binomial distribution:
μ = n * p
= 14 * 0.3
= 4.2
Step 2: Calculate the standard deviation (σ) of the binomial distribution:
σ = sqrt(n * p * (1 - p))
= sqrt(14 * 0.3 * 0.7)
= sqrt(2.94)
≈ 1.71
Step 3: Convert the binomial probability to a standard normal probability using a z-score:
z = (x - μ) / σ
= (5 - 4.2) / 1.71
≈ 0.47
Step 4: Look up the standard normal probability in a z-table or use a calculator to find the area to the left of the z-score. Let's assume the value is 0.679.
The normal approximation for the binomial probability that x=5 is approximately 0.679.
To compare this to the value of p(x=5) found in table 2, you would need to provide more information about the specific table you are referring to. The value of p(x=5) found in table 2 would represent the exact binomial probability, while the normal approximation is an estimate using the properties of the normal distribution.