Question

last year, more than 1.6 million students took the sat. the distribution of scores on the math section (out of 800) is approximately normal with a mean of 528 and standard deviation of 117. the university of michigan has a recommended sat math score of at least 730.
what percentage of students who took the sat math test meet this requirement? (round to 4 decimal places and then convert to a percentage.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 730\), \(\mu=528\), and \(\sigma = 117\).

$$z=\frac{730 - 528}{117}=\frac{202}{117}\approx1.7265$$

Step2: Find the probability using the standard normal distribution

We want to find \(P(X\geq730)\), which is equivalent to \(P(Z\geq1.7265)\) in the standard normal distribution. Since \(P(Z\geq z)=1 - P(Z < z)\), and looking up \(P(Z < 1.7265)\) in the standard normal table (or using a calculator with a normal - distribution function, e.g., for a TI - 84: normalcdf(-1000,1.7265,0,1)).
Using a calculator, \(P(Z < 1.7265)\approx0.9571\)
Then \(P(Z\geq1.7265)=1 - 0.9571 = 0.0429\)

Answer:

\(4.29\%\)