Question

the scores on a test are normally distributed with a mean of 50 and a standard deviation of 10. what is the score that is $2\frac{1}{2}$ standard deviations above the mean? a score of $\square$ is $2\frac{1}{2}$ standard deviations above the mean.

Explanation:

Step1: Convert the mixed number to an improper fraction

$2\frac{1}{2}=\frac{2\times2 + 1}{2}=\frac{5}{2}$

Step2: Use the formula for a value \(x\) in a normal distribution \(x=\mu+z\sigma\)

Here, \(\mu = 50\) (mean), \(z=\frac{5}{2}\) (number of standard deviations), \(\sigma = 10\) (standard deviation).
Substitute the values into the formula: \(x=50+\frac{5}{2}\times10\)
First calculate \(\frac{5}{2}\times10\): \(\frac{5\times10}{2}=25\)
Then \(x = 50+25\)

Answer:

\(75\)