Step 1 :We are given a normally distributed random variable $X$ with mean $\mu = 53$ and standard deviation $\sigma = 9$. We are asked to find the 15th percentile of this distribution.
Step 2 :The percentile of a value in a distribution is the percentage of values in the distribution that are less than that value. Therefore, the 15th percentile is the value such that 15% of the values are less than it.
Step 3 :We can find this value using the Z-score formula, which is $\frac{X - \mu}{\sigma}$, where $X$ is the value we're looking for, $\mu$ is the mean, and $\sigma$ is the standard deviation. However, in this case, we need to find $X$, so we rearrange the formula to $X = Z \cdot \sigma + \mu$.
Step 4 :The Z-score for the 15th percentile can be found from a standard normal distribution table. The Z-score for the 15th percentile is approximately -1.036.
Step 5 :Substituting the Z-score, the mean, and the standard deviation into the formula, we get $X = -1.036 \cdot 9 + 53$, which simplifies to $X = 43.67$.
Step 6 :Thus, the 15th percentile of the given normal distribution is $\boxed{43.67}$.