The cutoff score for the reading intervention program should be 58.5. This means that only students who score below 58.5 on the reading achievement exam will be eligible for the program.
How to solve
To determine the cutoff score for the reading intervention program, we'll first find the z-score that corresponds to the bottom 5% of students in the district.
Given the normally distributed reading scores, a z-score of -1.65 corresponds to the 5th percentile.
Using this z-score and the formula z = (x - μ) / σ, we can calculate the cutoff score (x) to be 58.5.
z = (x - μ) / σ
where:
z is the z-score (-1.65)
x is the cutoff score we want to find
μ is the average reading score (75)
σ is the standard deviation of reading scores (10)
Rearranging the formula to solve for x, we get:
x = μ + z * σ
Plugging in the values, we get:
x = 75 + (-1.65) * 10
x ≈ 58.5
Therefore, the cutoff score for the reading intervention program should be 58.5. This means that only students who score below 58.5 on the reading achievement exam will be eligible for the program.
The Complete Question
A school district is considering implementing a reading intervention program for students who score below a certain cutoff score on a reading achievement exam. If the average score for the students in the district is 75 and the standard deviation is 10, assume the variable is normally distributed. What cutoff score will make a student eligible for the program if only the bottom 5% of students should be eligible?