Question

the iq scores on the weschler adult intelligence scale (wais) follow a normal distribution with a mean of 100 and a standard deviation of 12. using standardized values, determine the iq scores that make up the middle 95% of iq scores. between 76 and 124 between 22 and 108 between 64 and 136 between 80 and 120

Explanation:

Step1: Recall the empirical rule for normal distributions

For a normal distribution, approximately 95% of the data lies within \(z = - 2\) and \(z=2\) (where \(z\) is the standardized value or z - score). The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation. We can re - arrange the formula to solve for \(x\): \(x=\mu + z\sigma\).

Step2: Calculate the lower bound

Given \(\mu = 100\), \(\sigma=12\), and \(z=-2\).
Substitute into the formula \(x=\mu + z\sigma\):
\(x_1=100+(-2)\times12\)
\(x_1 = 100-24\)
\(x_1 = 76\)

Step3: Calculate the upper bound

Given \(z = 2\), \(\mu = 100\), \(\sigma = 12\)
Substitute into the formula \(x=\mu+z\sigma\):
\(x_2=100 + 2\times12\)
\(x_2=100 + 24\)
\(x_2=124\)

Answer:

Between 76 and 124