15. Given the \( \sigma^{2}=16, x=57 \), and \( z=2.03 \). What is the valuo of \( \mu \) ?

To find the value of \( \mu \), we can use the z-score formula, which is: \[ z = \frac{x - \mu}{\sigma} \] Here, we can rearrange the formula to solve for \( \mu \): \[ \mu = x - z \cdot \sigma \] Given that \( \sigma^{2} = 16 \), we find \( \sigma = 4 \). Now, substituting the values: \( x = 57 \), \( z = 2.03 \), and \( \sigma = 4 \): \[ \mu = 57 - (2.03 \cdot 4) \] \[ \mu = 57 - 8.12 \] \[ \mu = 48.88 \] So, the value of \( \mu \) is approximately \( 48.88 \). To make the solution even clearer, z-scores provide a way to understand how far a particular observation (in this case, \( x = 57 \)) is from the mean (\( \mu \)) in terms of standard deviations. The z-score of 2.03 indicates that \( x \) is 2.03 standard deviations above the mean, making it a handy tool for assessing probabilities and thresholds in various contexts, like education and health assessments! Additionally, this concept is widely used in fields such as psychology and marketing research. For example, researchers often apply the normal distribution and z-scores to analyze test scores, where knowing how individuals perform relative to the average can help in decision-making processes such as admissions or targeted marketing strategies. It’s fascinating how these numbers influence diverse decisions!