To determine the minimum score needed to be in the top 14% of a normal distribution with a mean of 24 and a standard deviation of 3, we can follow these detailed steps:
1. Understand the Problem: We are dealing with a normal distribution, and we need to find the score that separates the top 14% from the rest. This score is also known as the percentile rank.
2. Find the Z-Score for the Top 14%:
- The top 14% means we are looking for the value that leaves 14% of the distribution to its right.
- In a standard normal distribution (mean = 0, standard deviation = 1), the z-score corresponding to the top 14% can be found using statistical tables or functions from statistical software.
3. Use the Z-Score Formula:
- For a normal distribution, the z-score formula is:
[tex]\[
Z = \frac{(X - \mu)}{\sigma}
\][/tex]
where [tex]\(Z\)[/tex] is the z-score, [tex]\(X\)[/tex] is the value we need to find, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
- Rearranging this formula to solve for [tex]\(X\)[/tex], we get:
[tex]\[
X = \mu + Z \cdot \sigma
\][/tex]
4. Apply the Given Values:
- The mean [tex]\( \mu \)[/tex] is 24.
- The standard deviation [tex]\( \sigma \)[/tex] is 3.
- The z-score corresponding to the top 14% is approximately 1.08. (This z-score tells us how many standard deviations the required score is above the mean.)
5. Calculate the Minimum Score:
- Plugging the values into the formula, we get:
[tex]\[
X = 24 + 1.08 \cdot 3
\][/tex]
[tex]\[
X \approx 24 + 3.24
\][/tex]
[tex]\[
X \approx 27.24
\][/tex]
6. Conclusion:
- The minimum score needed to be in the top 14% of this normal distribution is approximately 27.24.
So, the correct answer is:
- OC. X = 27.24
