Step 1 :To find the first quartile $Q_{1}$, we need to find the z-score that corresponds to the 25th percentile in a standard normal distribution.
Step 2 :The z-score is a measure of how many standard deviations an element is from the mean. In a standard normal distribution, the mean is 0 and the standard deviation is 1.
Step 3 :We can use a z-table or a calculator with a normal distribution function to find this z-score. The z-score corresponding to the 25th percentile is approximately -0.674.
Step 4 :Now, we can use the formula for the z-score to find the corresponding IQ score: \(Z = \frac{X - \mu}{\sigma}\)
Step 5 :Rearranging the formula to solve for X gives us: \(X = Z * \sigma + \mu\)
Step 6 :Substituting the given values gives us: \(X = -0.674 * 22 + 99.8\)
Step 7 :So, the first quartile $Q_{1}$, which is the IQ score separating the bottom 25% from the top 75%, is approximately \(\boxed{84.8}\)