High School Competency Test A mandatory competency test for high school sophomores has a normal distribution with a mean of 500 and a standard deviation of 106. Round the final answers to the nearest whole number and intermediate $z$-value calculations to 2 decimal places.

(b) The bottom $4 \%$ of students must go to summer school. What is the minimum score you would need to stay out of this group?
The minimum score needed to stay out of this group is $\square$.
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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.