By applying the Central Limit Theorem, the probability the sample's mean length is greater than 4.8 inches is 0.9868.
In Mathematics and Statistics, the z-score of a given sample size or data set can be calculated by using the following formula:
Z-score, z = (X - μ)/σ
Where:
- σ represents the standard deviation.
- X represents the sample score.
- μ represents the mean score.
In order to determine the standard deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size:
σx = σ/(√n)
σx = 0.5/(√32)
σx ≈ 0.09 inches.
Next, we would standardize the variable X by subtracting the mean and dividing by the standard deviation as follows;
Z-score, z = (4.8 - 5)/0.09
Z-score, z = -0.2/0.09
Z-score, z = -2.22
Based on the standardized normal distribution table, the required probability is given by:
P(X ≥ -2.22) = 1 - P(x < Z)
P(X ≥ -2.22) = 1 - 0.0132
Probability = 0.9868.
Percentage = 0.9868 × 100
Percentage = 98.68%.