Question-88
A random sample of size \(100\) is collected from a normal population with mean \(\mu\) and variance \(\sigma^{2}\). Suppose the expected value and variance of the sample mean is \(50\) and \(0.25\) respectively, find the value of \(\mu+10\sigma\).
\(100\)
Let the \(i^{\text{th}}\) observation in the random sample be \(X_{i} \sim N\left( \mu,\sigma^{2}\right)\), then the sample mean is:
\[ \overline{X} =\frac{X_{1} +\cdots +X_{100}}{100} \]
The expected value of the sample mean is just just \(\mu\) using the linearity of expectation and the i.i.d nature of the \(100\) random variables. The variance of \(\overline{X}\) is \(\frac{\sigma^{2}}{100}\) using the properties of variance of a sum of independent random variables. Therefore:
\[ \begin{aligned} \mu & =50\\ \frac{\sigma^{2}}{100} & =0.25 \end{aligned} \Longrightarrow \mu+10\sigma=100 \]