The scores on a test are normally distributed with a mean of 100 and a standard deviation of 30 . Find the score that is \( 3 \frac{1}{2} \) standard deviations above the mean. A score of \( \square \) is \( 3 \frac{1}{2} \) standard deviations above the mean.

To find the score that is \( 3 \frac{1}{2} \) standard deviations above the mean, we need to calculate it using the formula: \[ \text{Score} = \text{mean} + (z \times \text{standard deviation}) \] Here, the mean is 100, the standard deviation is 30, and \( z \) is \( 3.5 \). Plugging in the values: \[ \text{Score} = 100 + (3.5 \times 30) = 100 + 105 = 205. \] So, a score of \( 205 \) is \( 3 \frac{1}{2} \) standard deviations above the mean. --- Did you know that the concept of the normal distribution dates back to the 18th century, with the contributions of mathematicians like Pierre-Simon Laplace and Carl Friedrich Gauss? Gauss used it to model errors in astronomical observations, giving birth to the bell-shaped curve that we use today in statistics! In real-world applications, understanding how to interpret the normal distribution is crucial in fields like psychology and education, where test scores are analyzed. For instance, if you know the average test score and its variability, you can make informed decisions about curriculum adjustments, or even better target resources for student improvement based on where they fall in the distribution!