To find the number of students who scored less than 96 on the standardized test, we need to follow a detailed, step-by-step process. Let's walk through the solution:
1. Identify the given values:
- Mean ([tex]\(\mu\)[/tex]) = 156
- Standard deviation ([tex]\(\sigma\)[/tex]) = 23
- Total number of students ([tex]\(N\)[/tex]) = 48,592
- Score ([tex]\(X\)[/tex]) = 96
2. Calculate the z-score:
The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Substituting the provided values:
[tex]\[
z = \frac{96 - 156}{23} = \frac{-60}{23} \approx -2.6087
\][/tex]
3. Find the cumulative probability corresponding to the calculated z-score:
Use the z-table to find the cumulative probability for [tex]\(z \approx -2.6\)[/tex].
Looking at the provided z-table, we see:
- For [tex]\(z = -2.6\)[/tex], the cumulative probability is approximately 0.004.
- For [tex]\(z = -2.7\)[/tex], the cumulative probability is approximately 0.00347.
Interpolating, the cumulative probability for [tex]\(z \approx -2.6087\)[/tex] is approximately 0.00187.
4. Calculate the number of students who scored less than 96:
Multiply the total number of students by the cumulative probability.
[tex]\[
\text{Number of students} = N \times \text{cumulative probability}
\][/tex]
Substituting in the values:
[tex]\[
\text{Number of students} = 48,592 \times 0.00187 \approx 90.86704
\][/tex]
5. Select the correct answer:
Given the options and the calculated number of students (approximately 90.86704), the closest answer is:
C. 60
