To solve these questions, we'll need to use the properties of the normal distribution. The information provided suggests that you have a population with a mean (μx) of 45 and a standard deviation (σx) of 8.
To find P(EX > 2,400), you're looking for the probability that a sample mean (EX) is greater than 2,400. You need to calculate the z-score for 2,400, which is given by:
z = (X - μx) / σx
z = (2,400 - 45) / 8
Now, calculate this z-score and find the probability using a standard normal distribution table.
To find EX when z = -2, you can rearrange the z-score formula:
X = μx + (z * σx)
Substitute the given values:
X = 45 + (-2 * 8)
Calculate the value of X.
To find the 80th percentile for the sum of the 50 values of x, you can use the central limit theorem. The sum of a large sample from a population follows a normal distribution. First, calculate the mean (μ_sum) and standard deviation (σ_sum) of the sum of 50 values from the original population:
μ_sum = 50 * μx
σ_sum = √(50) * σx
Now, find the z-score for the 80th percentile (z_80) using a standard normal distribution table. Then, use the z-score and σ_sum to find the sum value (X_sum):
X_sum = μ_sum + (z_80 * σ_sum)
These calculations will help you find the answers to the three questions.
Explanation: