Step 1 :Given that we want to find the score that corresponds to the bottom 4% of students, we need to find the z-score that corresponds to a cumulative probability of 0.04 (4%).
Step 2 :Using a standard normal distribution table or a z-score calculator, we find that the z-score corresponding to a cumulative probability of 0.04 is approximately -1.75.
Step 3 :We can use the formula for a z-score to find the corresponding test score: \(Z = (X - μ) / σ\), where Z is the z-score, X is the value we're trying to find, μ is the mean, and σ is the standard deviation.
Step 4 :Rearranging the formula to solve for X gives us: \(X = Z * σ + μ\).
Step 5 :Substituting the given values into the formula gives us: \(X = -1.75 * 106 + 500\).
Step 6 :Calculating this gives us: \(X = -185 + 500 = 315\).
Step 7 :So, the minimum score needed to stay out of the bottom 4% of students is approximately 315. This is the score you would need to avoid going to summer school.
Step 8 :Checking our answer by substituting \(X = 315\) back into the z-score formula, we get: \(Z = (315 - 500) / 106 = -1.75\). This is the z-score we started with, so our answer is correct.
Step 9 :\(\boxed{315}\) is the minimum score needed to stay out of the bottom 4% of students.