Question

on a standardized test with normal distribution, the mean is 75 and the standard deviation is 6. if 1200 students took the test, approximately how many students would be expected to score between 69 and 81?
1140
408
600
816

Explanation:

Step1: Calculate the number of standard deviations from the mean

The formula for the number of standard deviations \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 75\) (mean), \(\sigma=6\) (standard deviation).
For \(x = 69\): \(z_1=\frac{69 - 75}{6}=\frac{- 6}{6}=-1\)
For \(x = 81\): \(z_2=\frac{81 - 75}{6}=\frac{6}{6}=1\)

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution, approximately \(P(-1<Z<1)=0.68\) (where \(Z\) is the standard - normal variable)

Step3: Calculate the number of students

If \(N = 1200\) (total number of students), then the number of students with scores between \(69\) and \(81\) is \(n=N\times P(-1 < Z<1)\)
Substitute \(N = 1200\) and \(P(-1<Z<1)=0.68\) into the formula: \(n = 1200\times0.68=816\)

Answer:

816