Answer:
The probability that fewer than 22% of the sample will be comprised of first-generation students is approximately 0.0283, or 2.83%.
Explanation:
To estimate the probability that fewer than 22% of the sample will be comprised of first-generation students in an independent sample of 79 Linfield students, you can use the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample proportion approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, as long as the sample size is sufficiently large.
Here's how you can calculate this probability:
Find the mean (μ) and standard deviation (σ) of the sample proportion:
μ = p = 0.32 (proportion of first-generation students in the population)
σ = sqrt[(p * (1 - p)) / n]
Where n is the sample size (79).
μ = 0.32
σ = sqrt[(0.32 * (1 - 0.32)) / 79]
σ ≈ sqrt[(0.2176) / 79]
σ ≈ sqrt(0.002755)
σ ≈ 0.0525 (rounded to four decimal places)
Calculate the z-score for the sample proportion of 0.22 (22%):
z = (X - μ) / σ
z = (0.22 - 0.32) / 0.0525
z ≈ (-0.1) / 0.0525
z ≈ -1.9048 (rounded to four decimal places)
Look up the z-score in a standard normal distribution table or use a calculator to find the corresponding cumulative probability (P(Z < -1.9048)).
P(Z < -1.9048) ≈ 0.0283
So, the probability that fewer than 22% of the sample will be comprised of first-generation students is approximately 0.0283, or 2.83%.