Questions: Suppose 47% of the population has a college degree. If a random sample of size 460 is selected, what is the probability that the proportion of persons with a college degree will differ from the population proportion by less than 5%?

Solution Steps

The population proportion is given as _p_ = 0.47 and the sample size is _n_ = 460. The standard deviation of the sample proportion (σ_p̂) is calculated as follows:

σ_p̂ = sqrt([_p_(1 - _p_)]/_n_) = sqrt([0.47 * (1 - 0.47)]/460) = sqrt(0.0005254) ≈ 0.023

We want to find the probability that the sample proportion (p̂) is within 0.05 of the population proportion. This can be expressed as:

P(0.47 - 0.05 < p̂ < 0.47 + 0.05) = P(0.42 < p̂ < 0.52)

We convert these bounds to z-scores using the formula _z_ = (p̂ - _p_) / σ_p̂:

_z_₁ = (0.42 - 0.47)/0.023 ≈ -2.17 _z_₂ = (0.52 - 0.47)/0.023 ≈ 2.17

Therefore, we are looking for P(-2.17 < _z_ < 2.17)

We can find the probability corresponding to the calculated z-scores using a standard normal distribution table (z-table). P(_z_ < 2.17) ≈ 0.9850 P(_z_ < -2.17) ≈ 0.0150 P(-2.17 < _z_ < 2.17) = P(_z_ < 2.17) - P(_z_ < -2.17) ≈ 0.9850 - 0.0150 ≈ 0.97

Final Answer: The probability that the sample proportion of college degrees will differ from the population proportion by less than 5% is approximately 0.97.