Question

a standard iq test produces normally distributed results with a mean of 104 and a standard deviation of 16 for 52,000 students in grade 12. approximately how many of these students would have iqs below 76? 2080 23,915 4170 49,915

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 76\), \(\mu=104\), and \(\sigma = 16\).

$$z=\frac{76 - 104}{16}=\frac{-28}{16}=-1.75$$

Step2: Find the proportion using the standard normal distribution table

Looking up the z - score of \(-1.75\) in the standard normal distribution table (or using a calculator with a normalcdf function: normalcdf\((-\infty,-1.75)\)), we find that the proportion of values below \(z=-1.75\) is approximately \(0.0401\).

Step3: Calculate the number of students

Multiply the proportion by the total number of students \(n = 52000\).

$$N=0.0401\times52000 = 2085.2\approx2080$$

Answer:

2080