Z score transformation

The annual salary of an entry-level statistics major (in thousands of dollars) is normally distributed with a mean of 75 and a standard deviation of 12. X ∼ N ( μ = 75, σ = 12 ). What is the minimum salary a statistics major should aim for in order to earn amongst the top 10%?

Correct answer:

m₁₀ =  90.36

Step-by-step explanation:

$$\begin{gathered} \mu =75 \\ \sigma =12 \\ X\dots N(\mu =75,\sigma =12) \\ {p}_{10}=100\%-10\%=1-\frac{10}{100}=0.9 \\ p(z<z_{10})=1-0.1=0.9 \\ {z}_{10}=1.28 \\ {z}_{10}=\frac{m_{10}-\mu }{\sigma } \\ {m}_{10}=z_{10}\cdot \sigma +\mu =1.28\cdot 12+75=\frac{2259}{25}=90\frac{9}{25}=90.36 \end{gathered}$$

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