Concept
Apply z scores
First, let's find the z-score that corresponds to the information given for Exam A.
[tex]\begin{gathered} \text{Mean }\mu\text{ = 200} \\ \text{Standard debiation }\sigma\text{ = 20} \\ z\text{ = }\frac{x\text{ - }\mu}{\sigma} \\ \text{Substitute the following values} \\ x\text{ = 2}16 \\ \sigma\text{ = 20} \\ \mu\text{ = 200} \\ z\text{ = }\frac{216\text{ - 200}}{20} \\ z\text{ = }\frac{16}{20} \\ z\text{ = 0.8} \end{gathered}[/tex]For Sophie to do equivalently well in Exam B as well as she did on Exam A, we find the value of x by using the z score of exam A.
The formula for finding a z-score is shown below:
[tex]\begin{gathered} \text{Exam B} \\ x\text{ = ?} \\ z\text{ score = 0.8} \\ \mu\text{ = 450} \\ \sigma\text{ = 40} \end{gathered}[/tex]Next, substitute the following values to find the value of x.
[tex]\begin{gathered} \text{Therefore,} \\ z\text{ = }\frac{x-\text{ }\mu}{\sigma} \\ 0.8\text{ = }\frac{x\text{ - 450}}{40} \\ \text{Cross multiply} \\ x\text{ - 450 = 0.8 x 40} \\ x\text{ - 450 = 32} \\ \text{x = 450 + 32} \\ x\text{ = 482} \end{gathered}[/tex]
Final answer
x = 482
