Question

iq scores have roughly a normal distribution with a mean, \\( \mu = 98 \\) and standard deviation, \\( \sigma = 11 \\). what proportion of people have an iq score less than 112? \\( p ( x < 112 ) = \\) which is the correct graph that corresponds to the above percentage?

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 98\), \(\sigma=11\), and \(x = 112\).

$$z=\frac{112 - 98}{11}=\frac{14}{11}\approx1.27$$

Step2: Find the probability using the standard normal table

We want to find \(P(X\lt112)\), which is equivalent to \(P(Z\lt1.27)\) in the standard normal distribution (\(Z=\frac{X-\mu}{\sigma}\)).
Looking up the value of \(z = 1.27\) in the standard normal table (the cumulative - distribution function of the standard normal distribution \(\varPhi(z)\)).
From the standard normal table, \(P(Z\lt1.27)=0.8980\)

Answer:

\(0.8980\)