Answer :
To solve the problem of determining the maximum SAT score in 2014 that meets the test-retake preparation course requirements, we need to find the score that represents the 25th percentile of the SAT score distribution. Here are the steps to reach the solution:
1. Understand the Given Data:
- The distribution of SAT scores is approximately normal.
- Mean SAT score ([tex]$\mu$[/tex]) = 1497
- Standard deviation ([tex]$\sigma$[/tex]) = 322
- We need to determine the SAT score that is at the 25th percentile, i.e., the score below which 25% of the scores fall.
2. Z-Score Calculation:
The 25th percentile in a normal distribution corresponds to a specific z-score, which is a standardized score that tells us how many standard deviations a value is from the mean. This value can be found using statistical tables or tools for the standard normal distribution.
For the 25th percentile, the z-score is:
[tex]\[
z \approx -0.6744897501960817
\][/tex]
3. Convert the Z-Score to an SAT Score:
To find the actual SAT score corresponding to this z-score, we use the following formula:
[tex]\[
\text{SAT score} = \mu + z \cdot \sigma
\][/tex]
Plugging in the values:
[tex]\[
\text{SAT score} = 1497 + (-0.6744897501960817 \cdot 322)
\][/tex]
4. Perform the Calculation:
[tex]\[
\text{SAT score} \approx 1497 - 217.18569956313832
\][/tex]
[tex]\[
\text{SAT score} \approx 1279.8143004368617
\][/tex]
Therefore, the maximum SAT score in 2014 that qualifies for the test-retake preparation course, which targets students in the lower 25%, is approximately 1280.
1. Understand the Given Data:
- The distribution of SAT scores is approximately normal.
- Mean SAT score ([tex]$\mu$[/tex]) = 1497
- Standard deviation ([tex]$\sigma$[/tex]) = 322
- We need to determine the SAT score that is at the 25th percentile, i.e., the score below which 25% of the scores fall.
2. Z-Score Calculation:
The 25th percentile in a normal distribution corresponds to a specific z-score, which is a standardized score that tells us how many standard deviations a value is from the mean. This value can be found using statistical tables or tools for the standard normal distribution.
For the 25th percentile, the z-score is:
[tex]\[
z \approx -0.6744897501960817
\][/tex]
3. Convert the Z-Score to an SAT Score:
To find the actual SAT score corresponding to this z-score, we use the following formula:
[tex]\[
\text{SAT score} = \mu + z \cdot \sigma
\][/tex]
Plugging in the values:
[tex]\[
\text{SAT score} = 1497 + (-0.6744897501960817 \cdot 322)
\][/tex]
4. Perform the Calculation:
[tex]\[
\text{SAT score} \approx 1497 - 217.18569956313832
\][/tex]
[tex]\[
\text{SAT score} \approx 1279.8143004368617
\][/tex]
Therefore, the maximum SAT score in 2014 that qualifies for the test-retake preparation course, which targets students in the lower 25%, is approximately 1280.
