(CH $5 \& 6$ )
Question 15 of 19
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Assume that adults have IQ scores that are normally distributed with a mean of 99.8 and a standard deviation 22 . Find the first quartile $Q_{1}$, which is the IQ score separating the bottom $25 \%$ from the top $75 \%$. (Hint: Draw a graph.) :
The first quartile is
(Type an integer or decimal rounded to one decimal place as needed.)

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 \(\boxed{84.8}\)