Question

Given that the scores of a set of candidates on an IQ test are normally distributed, if the IQ test has a mean of 100 and a standard deviation of 10, determine the probability that a candidate who takes the test will score between 90 and 110.
[Given \( P(Z<1) = 0.8413 \) and \( P(Z<-1) = 0.1587 \)]

Hint:

For normally distributed data, standardize using \( Z = \frac{X - \mu}{\sigma} \), and use the standard normal table to find probabilities.

Answer 1

Step 1: Standardize the scores using \( Z = \frac{X - \mu}{\sigma} \): For \( X = 90 \): \[ Z = \frac{90 - 100}{10} = -1. \] For \( X = 110 \): \[ Z = \frac{110 - 100}{10} = 1. \] Step 2: Use the standard normal table: \[ P(90 \leq X \leq 110) = P(Z<1) - P(Z<-1). \] Substitute values: \[ P(90 \leq X \leq 110) = 0.8413 - 0.1587 = 0.6826. \] Thus, the probability is \( 0.6826 \) or \( 68.26\% \).