Answer:
Mean = 26
Standard deviation = √2
Explanation:
If we have two independent variables X and Y, we can use the following properties:
[tex]\begin{gathered} \mu(X+Y)=\mu(X)+\mu(Y) \\ \mu(aX)=a\cdot\mu(X) \\ \sigma^2(X+Y)=\sigma^2(X)+\sigma^2(Y) \\ \sigma^2(aX)=a^2\cdot\sigma^2(X) \end{gathered}[/tex]Where μ is the means and σ² is the variance of the variables.
Now, we can apply these properties to calculate the mean of the variable X hat as:
[tex]\begin{gathered} \bar{X}=\frac{X_1+X_2_{}}{2}_{} \\ \mu(\bar{X})=\mu(\frac{X_1+X_2}{2}) \\ \mu(\bar{X})=\frac{1}{2}\mu(X_1+X_2) \\ \mu(\bar{X})=\frac{1}{2}(\mu(X_1)+\mu(X_2)) \end{gathered}[/tex]Since μ(X1) = 26 and μ(X2) = 26, μ(Xhat) is equal to:
[tex]\begin{gathered} \mu(\bar{X})=\frac{1}{2}(26+26) \\ \mu(\bar{X})=26 \end{gathered}[/tex]Now, to find the standard deviation, we need to find first the variance of X hat as:
[tex]\begin{gathered} \sigma^2(\bar{X})=\sigma^2(\frac{X_1+X_2}{2}) \\ \sigma^2(\bar{X})=(\frac{1}{2})^2\cdot\sigma^2(X_1+X_2) \\ \sigma^2(\bar{X})=\frac{1}{4}\cdot(\sigma^2(X_1)+\sigma^2(X_2)) \end{gathered}[/tex]The variance is equal to the square of the standard deviation, so replacing σ²(X1) by 4 and σ²(X2) by 4, we get:
[tex]\begin{gathered} \sigma^2(\bar{X})=\frac{1}{4}(4+4) \\ \sigma^2(\bar{X})=2 \end{gathered}[/tex]Therefore, the standard deviation is equal to:
[tex]\sigma(\bar{X})=\sqrt[]{2}[/tex]So, the mean is 26 and the standard deviation is √2
