To determine how many of the 48,592 students scored less than 96 on the standardized test, we need to follow these steps:
1. Find the z-score:
The z-score tells us how many standard deviations a particular score is from the mean. The formula to calculate the z-score is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\( X \)[/tex] is the score of interest (96 in this case).
- [tex]\( \mu \)[/tex] is the mean (156).
- [tex]\( \sigma \)[/tex] is the standard deviation (23).
Plugging in the given values:
[tex]\[
z = \frac{96 - 156}{23} = \frac{-60}{23} \approx -2.6087
\][/tex]
2. Look up the cumulative probability for the z-score [tex]\( -2.6087 \)[/tex]:
We need to look up the cumulative probability from the z-table, which tells us the probability that a score is less than a given z-score. Checking the z-table for [tex]\( z \approx -2.61 \)[/tex] (rounding [tex]\( -2.6087 \)[/tex] to [tex]\( -2.61 \)[/tex]), we get a cumulative probability approximately [tex]\( 0.00454 \)[/tex].
3. Calculate the number of students below the given score:
To find the number of students who scored less than 96:
[tex]\[
\text{Number of students} = \text{cumulative probability} \times \text{total number of students}
\][/tex]
Plugging in the values:
[tex]\[
\text{Number of students} = 0.00454 \times 48592 \approx 220.82
\][/tex]
Since the number of students must be a whole number, we approximate to the nearest whole number. Thus, about 221 students scored less than 96.
Therefore, the number of students who scored less than 96 is approximately:
[tex]\[
\boxed{220}
\][/tex]
