To find the score that is [tex]\(\frac{1}{2}\)[/tex] standard deviation above the mean in a normally distributed set of test scores, we can follow these steps:
1. Identify the mean and the standard deviation:
- The mean ([tex]\(\mu\)[/tex]) is 100.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 20.
2. Determine the fraction of the standard deviation above the mean:
- We need to find the value that is [tex]\(\frac{1}{2}\)[/tex] (or 0.5) standard deviations above the mean.
3. Calculate the score:
- To find this score, we calculate the product of the standard deviation and the desired multiple, then add it to the mean.
- [tex]\[\text{Score} = \mu + (\sigma \times 0.5)\][/tex]
4. Perform the arithmetic:
- [tex]\(\sigma \times 0.5 = 20 \times 0.5 = 10\)[/tex]
- Add this value to the mean:
[tex]\[\text{Score} = 100 + 10 = 110\][/tex]
Therefore, the score that is [tex]\(\frac{1}{2}\)[/tex] standard deviation above the mean is 110.0.
