1. P(X ≤ 0.35):
For a normal distribution, we can calculate the z-score of X:
z = (X - μ) / σ = (0.35 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 0.71
Using a standard normal distribution table or calculator, we find the area to the left of z = 0.71:
P(X ≤ 0.35) ≈ 0.7611
2. P(X > 0.40):
Calculate the z-score of :X
z = (0.40 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 1.41
Using a standard normal distribution table or calculator, we find the area to the right of z = 1.41:
P(X > 0.40) ≈ 0.0783
3. P(0.25 < X < 0.40):
Calculate the z-scores of the boundaries:
z1 = (0.25 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ -0.71
z2 = (0.40 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 1.41
Using a standard normal distribution table or calculator, find the area between z = -0.71 and z = 1.41:
P(0.25 < X < 0.40) ≈ 0.8124 - 0.2611 ≈ 0.5513
4. P(X ≤ 0.27):
Calculate the z-score of X:
z = (0.27 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ -0.45
Using a standard normal distribution table or calculator, find the area to the left of z = -0.45:
P(X ≤ 0.27) ≈ 0.3228
The Complete Question
A population has a proportion equal to 0.30. Calculate the following probabilities with n = 100:
P(X ≤ 0.35)
P(X > 0.40)
P(0.25 < X < 0.40)
P(X ≤ 0.27)
Additionally, assume the population follows a normal distribution.