Question

the scores on a test are normally distributed with a mean of 30 and a standard deviation of 6. what is the score that is 2\frac{1}{2} standard deviations above the mean? a score of \boxed{} is 2\frac{1}{2} standard deviations above the mean.

Explanation:

Step1: Recall the formula for a value in a normal distribution

To find a score that is a certain number of standard deviations above the mean, we use the formula: \( \text{Score} = \mu + z \times \sigma \), where \( \mu \) is the mean, \( z \) is the number of standard deviations, and \( \sigma \) is the standard deviation.

Step2: Identify the given values

We know that \( \mu = 30 \), \( \sigma = 6 \), and \( z = 2\frac{1}{2} = \frac{5}{2} = 2.5 \).

Step3: Substitute the values into the formula

Substitute \( \mu = 30 \), \( z = 2.5 \), and \( \sigma = 6 \) into the formula:
\( \text{Score} = 30 + 2.5 \times 6 \)

Step4: Calculate the result

First, calculate \( 2.5 \times 6 = 15 \). Then, add this to the mean: \( 30 + 15 = 45 \).

Answer:

45