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Assume X and Y are indepndent. Assume that X has mean 10 and standard deviation 8. Assume Y has mean 12 and standard deviation 6. What is the standard deviation of the variable Z = X + Y. a. 8 b. 6 c. 10 d. 12

Answer :

MrRoyal

Answer:

[tex]\sigma_z = 10[/tex]

Explanation:

Given

[tex]\bar x = 10[/tex]

[tex]\sigma_x = 8[/tex]

[tex]\bar y = 12[/tex]

[tex]\sigma_y = 6[/tex]

[tex]z = x + y[/tex]

Required

[tex]\sigma_z[/tex]

This is calculated as:

[tex]\sigma_z = \sqrt{Var(x) + Var(y)}[/tex]

Calculate [tex]Var(x)[/tex]

We have:

[tex]Var(x) = \sigma_x^2[/tex]

[tex]Var(x) = 8^2[/tex]

[tex]Var(x) = 64[/tex]

Calculate [tex]Var(y)[/tex]

We have:

[tex]Var(y) = \sigma_y^2[/tex]

[tex]Var(y) = 6^2[/tex]

[tex]Var(y) = 36[/tex]

So, we have:

[tex]\sigma_z = \sqrt{Var(x) + Var(y)}[/tex]

[tex]\sigma_z = \sqrt{64 + 36}[/tex]

[tex]\sigma_z = \sqrt{100}[/tex]

[tex]\sigma_z = 10[/tex]

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