\begin{tabular}{l} 9:37 \\ 3.11 Auto insurance premiums: Suppose a \\ newspaper article states that the distribution of \\ auto insurance premiums for residents of California \\ is approximately normal with a mean of \( \$ 1650 \). The \\ article also states that \( 25 \% \) of California residents \\ pay more than \( \$ 1900 \). \\ d) Identify the standard deviation of insurance \\ premiums in California. (Keep 1 decimal place) \\ \hline 409.8 \\ Submit All Parts \\ 4 \end{tabular}

To determine the standard deviation of insurance premiums in California, we can utilize the properties of the normal distribution. Given that 25% of residents pay more than $1900, this implies that $1900 corresponds to the 75th percentile of the distribution. Using the z-score formula, we can find the z-value associated with the 75th percentile, which is approximately 0.674. Now, we can set up the equation based on the relationship between the mean, z-score, and standard deviation: \[ Z = \frac{X - \mu}{\sigma} \] Substituting the values we have: \[ 0.674 = \frac{1900 - 1650}{\sigma} \] From this, we can solve for the standard deviation \(\sigma\): \[ 0.674\sigma = 250 \] \[ \sigma = \frac{250}{0.674} \approx 370.5 \] Therefore, the standard deviation of the insurance premiums in California is approximately \$370.5.