Estimators, standard error and biasEdexcel A-Level Further Maths: Mind map

Exam questions on Estimators, standard error and bias

1. $X_1$, $X_2$, $X_3$ is a random sample from a population with mean $\mu$ and variance $\sigma^2$. The statistic $U=\frac{X_1+2X_2+3X_3}{6}$ is proposed as an estimator of $\mu$.
   The sample mean $\bar X=\frac{X_1+X_2+X_3}{3}$ is also an estimator of $\mu$. Compare $U$ and $\bar X$ and state, with a reason, which you would use.2 marks
2. A random sample of $10$ observations of a quantity $x$ from a population with unknown mean and variance gives $\sum x=124.0$ and $\sum x^2=1582.6$.
   A larger sample of $40$ observations has the same sample variance as in part (a). Calculate the estimated standard error of its mean and state what this shows about $\bar X$ as an estimator of the population mean.2 marks
3. $X_1,X_2,\ldots,X_n$ is a random sample from a population with mean $\mu$ and variance $\sigma^2$. Two estimators of $\sigma^2$ are $S^2=\frac{1}{n-1}\sum(X_i-\bar X)^2$, which is known to be unbiased, and $V=\frac{1}{n}\sum(X_i-\bar X)^2$.
   Show that $V$ is a biased estimator of $\sigma^2$, and state the bias.3 marks