Assume that the population of heights of male college students is approximately normally distributed with mean m of 70.63 inches and standard deviation s of 6.47 inches. A random sample of 88 heights is obtained. Show all work.
(A) Find the mean and standard error of the x bar distribution
(B) Find P( > 72.25)
(A) To find the mean and standard error of the x bar distribution, we need to use the formulas:
Mean of x bar distribution (μx̄) = μ (mean of the population)
Standard Error of x bar distribution (σx̄) = σ / √n (standard deviation of the population divided by the square root of the sample size)
Given:
μ (population mean) = 70.63 inches
σ (population standard deviation) = 6.47 inches
n (sample size) = 88
Calculating the mean of the x bar distribution:
μx̄ = μ = 70.63 inches
Calculating the standard error of the x bar distribution:
σx̄ = σ / √n
σx̄ = 6.47 / √88
σx̄ ≈ 0.6904 inches
Therefore, the mean (μx̄) of the x bar distribution is 70.63 inches and the standard error (σx̄) is approximately 0.6904 inches.
(B) To find P(X > 72.25), where X is a randomly selected height from the given population, we need to use the z-score formula and the standard normal distribution table.
The formula for the z-score is:
z = (X - μ) / σ
Given:
X = 72.25 inches (the value we want to find the probability for)
μ = 70.63 inches (mean of the population)
σ = 6.47 inches (standard deviation of the population)
Calculating the z-score:
z = (72.25 - 70.63) / 6.47
z ≈ 0.2504
Using the standard normal distribution table (z-table) or a calculator with a cumulative distribution function (CDF) for the standard normal distribution, we can find the probability P(Z > 0.2504).
P(Z > 0.2504) can be calculated by subtracting the cumulative probability from 0.5:
P(Z > 0.2504) = 1 - P(Z ≤ 0.2504)
By referring to the z-table or using a calculator, we find that P(Z ≤ 0.2504) is approximately 0.5987.
Therefore,
P(Z > 0.2504) ≈ 1 - 0.5987
P(Z > 0.2504) ≈ 0.4013
So, the probability of randomly selecting a height greater than 72.25 inches from this population is approximately 0.4013.