To determine how many students scored less than 96 on this standardized test, let's follow these steps systematically:
1. Determine the z-score:
- The z-score formula is given by:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where [tex]\( x \)[/tex] is the score to find (96), [tex]\( \mu \)[/tex] is the mean score (156), and [tex]\( \sigma \)[/tex] is the standard deviation (23).
- Plugging in the values:
[tex]\[
z = \frac{96 - 156}{23} = \frac{-60}{23} \approx -2.6087
\][/tex]
2. Find the cumulative probability associated with the z-score:
- Using the provided z-score table, we observe that our exact z-score (-2.6087) is not available in the table. However, we can determine its probability through interpolation using nearby values.
- From the table:
[tex]\[
\begin{aligned}
\text{For } z = -2.6: & \quad \text{Probability} = 0.00403 \\
\text{For } z = -2.7: & \quad \text{Probability} = 0.00326 \\
\end{aligned}
\][/tex]
- We need a value between [tex]\(-2.6\)[/tex] and [tex]\(-2.7\)[/tex]. Interpolation can be done as:
[tex]\[
\text{Interpolated probability for } z = -2.6087 \approx 0.003169
\][/tex]
3. Calculate the number of students corresponding to this cumulative probability:
- The cumulative probability tells us the proportion of students scoring less than a particular value.
- Multiply this probability by the total number of students (48,592):
[tex]\[
\text{Number of students} = 0.003169 \times 48,592 \approx 154
\][/tex]
So, approximately 154 students scored less than 96 on this standardized test.
