Question

the scores on a test are normally distributed with a mean of 100 and a standard deviation of 30. find 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: Recall the formula for a value in a normal distribution

The formula to find a value \( x \) that is \( z \) standard deviations above the mean \( \mu \) is \( x=\mu + z\sigma \), where \( \mu \) is the mean, \( \sigma \) is the standard deviation, and \( z \) is the number of standard deviations.
Here, \( \mu = 100 \), \( \sigma = 30 \), and \( z = 2\frac{1}{2}=\frac{5}{2}=2.5 \).

Step2: Substitute the values into the formula

Substitute \( \mu = 100 \), \( \sigma = 30 \), and \( z = 2.5 \) into the formula \( x=\mu + z\sigma \).
\( x = 100+2.5\times30 \)

Step3: Calculate the result

First, calculate \( 2.5\times30 = 75 \). Then, add this to the mean: \( 100 + 75=175 \).

Answer:

175