Respuesta :
To check how many students scored between this range, we need to calculate the area of the distribution inside this range. To do that, we're going to use a z-table.
A z-table is a table with the correspondence between the z-score and the area below the graph. To use a z-table, first we need to convert those student scores to z-scores, and we do that using the following formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]From the question text, we have the following information:
[tex]\begin{gathered} \mu=82.3 \\ \sigma=3.5 \\ x_1=80 \\ x_2=90 \end{gathered}[/tex]Turning this into z-scores, we have:
[tex]\begin{gathered} z_1=\frac{(80-82.3)}{3.5}\approx-0.66 \\ z_2=\frac{(90-82.3)}{3.5}=2.2 \end{gathered}[/tex]For each z-score, we have the following area:
[tex]\begin{gathered} A_1=0.2454 \\ A_2=0.4861 \end{gathered}[/tex]If you sum them up, you're going to get the percentage of students between this range:
[tex]A_1+A_2=0.7315[/tex]Now, we know 73.15% of the students had a score between 80 and 90. Since the total amount of students is 78 we just calculate the percentage out of this value:
[tex]\frac{73.15}{100}\times78\approx57[/tex]Then, we have our answer.
Approximately 57 students scored between an 80 and a 90
