Answer :
The actual SAT score corresponding to the 47th percentile is approximately 1125.75.
To find the actual SAT score corresponding to the 47th percentile, we need to use the formula for the inverse cumulative distribution function (CDF) of the normal distribution, also known as the Z-score formula.
The formula to find the Z-score is given by:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Where:
- Z is the Z-score,
- X is the value we're interested in (the actual SAT score in this case),
- [tex]\( \mu \)[/tex] is the mean of the distribution (1101 for SAT scores),
- [tex]\( \sigma \)[/tex] is the standard deviation of the distribution (198 for SAT scores).
We are given that the percentile corresponds to the 47th percentile, which means that 47% of the scores fall below the given score.
Using a Z-table or a calculator to find the Z-score corresponding to the 47th percentile (which is approximately 0.1251), we can then rearrange the Z-score formula to solve for X:
[tex]\[ X = Z \times \sigma + \mu \][/tex]
Substituting the given values:
[tex]\[ X = 0.1251 \times 198 + 1101 \][/tex]
[tex]\[ X \approx 1125.7498 \][/tex]
So, the actual SAT score corresponding to the 47th percentile is approximately 1125.75.
