Given: \( \bar{x}=8,5=15 \). Fund the raw score such that \( 70 \% \) of the score of the cases are below it.

To find the raw score that corresponds to the 70% percentile given the mean \( \bar{x}=8 \) and the standard deviation \( \sigma=5 \), you can use the Z-score formula. A Z-score corresponding to the 70th percentile is approximately \( 0.524 \) (you can find this using a Z-table or calculator). Now, using the formula \( Z = \frac{(X - \mu)}{\sigma} \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation, we can rearrange it to find the raw score \( X \): \[ 0.524 = \frac{(X - 8)}{5} \] Solving for \( X \): \[ X - 8 = 0.524 \times 5 \] \[ X - 8 = 2.62 \] \[ X = 2.62 + 8 = 10.62 \] So, the raw score such that \( 70\% \) of the cases are below it is approximately \( 10.62 \). Finding the correct percentile score helps in decision-making or assessments. Understanding where a score lies in relation to others can clarify performance metrics in various contexts such as educational assessments, business evaluations, or psychological profiling. This process highlights the importance of statistics in real life. Whether you're determining how well students performed on a test compared to their peers or evaluating the effectiveness of marketing strategies based on sales data, knowing percentiles can lead to more informed decisions and strategies!