Sure! Let's find the score that is [tex]\(2 \frac{1}{2}\)[/tex] standard deviations above the mean, given a normal distribution with a mean ([tex]\(\mu\)[/tex]) of 150 and a standard deviation ([tex]\(\sigma\)[/tex]) of 30.
1. Identify the given values:
- Mean, [tex]\(\mu = 150\)[/tex]
- Standard deviation, [tex]\(\sigma = 30\)[/tex]
- Number of standard deviations above the mean, [tex]\( z = 2 \frac{1}{2} \)[/tex] (which is 2.5 in decimal form)
2. Understand the relationship between the mean, standard deviation, and the score:
The score ([tex]\(X\)[/tex]) that is [tex]\(z\)[/tex] standard deviations above the mean can be found using the formula:
[tex]\[
X = \mu + z\sigma
\][/tex]
Here, [tex]\(\mu\)[/tex] is the mean, [tex]\(z\)[/tex] is the number of standard deviations, and [tex]\(\sigma\)[/tex] is the standard deviation.
3. Substitute the values into the formula:
[tex]\[
X = 150 + 2.5 \times 30
\][/tex]
4. Calculate the product of [tex]\(2.5\)[/tex] and [tex]\(30\)[/tex]:
[tex]\[
2.5 \times 30 = 75
\][/tex]
5. Add this product to the mean:
[tex]\[
X = 150 + 75 = 225
\][/tex]
Therefore, a score of [tex]\(225\)[/tex] is [tex]\(2 \frac{1}{2}\)[/tex] standard deviations above the mean.
