Estimators, standard error and biasEdexcel A-Level Further Maths: Subtopic test

1. 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 $\mu$.

   (a)

   Which statement about $U$ is correct?

   [1 mark]

   • A$U$ is unbiased, because $\mathrm{E}(U)=\mu$
   • B$U$ is biased, because the weights are not all equal
   • C$U$ is biased, because $\mathrm{E}(U)=6\mu$
   • D$U$ is unbiased only if $\mu=0$

   (b)

   Find $\mathrm{Var}(U)$.

   [1 mark]

   • A$\frac{\sigma^2}{3}$
   • B$\frac{\sigma^2}{6}$
   • C$\frac{7\sigma^2}{3}$
   • D$\frac{7\sigma^2}{18}$

   (c)

   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]

   Total for question 1: 4 marks

2. 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)

   Calculate an unbiased estimate of the population variance.

   [1 mark]

   • A$4.5$
   • B$5$
   • C$2.24$
   • D$45$

   (b)

   Calculate the estimated standard error of the sample mean.

   [1 mark]

   • A$0.224$
   • B$2.24$
   • C$0.707$
   • D$0.5$

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $V=\frac{1}{n}\sum(X_i-\bar X)^2$.

   (a)

   Show that $V$ is a biased estimator of $\sigma^2$, and state the bias.

   [3 marks]

   (b)

   A sample of size $8$ has $\sum(x_i-\bar x)^2=56$. Calculate the estimates of $\sigma^2$ given by $S^2$ and by $V$. Hence state, as a percentage of $\sigma^2$, the size of the bias of $V$ for $n=8$, and say which estimator you would use.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $X_1,X_2,X_3,X_4$ is a random sample from a population with mean $\mu>0$ and variance $\sigma^2$. Three estimators of $\mu$ are $A=\frac{X_1+X_2+X_3+X_4}{4}$, $B=\frac{X_1+X_2}{2}$ and $C=\frac{X_1+X_2+X_3+X_4}{5}$.

   (a)

   (i) Show that $A$ is an unbiased estimator of $\mu$.
   (ii) Show that $B$ is unbiased.
   (iii) Find $\mathrm{E}(C)$ and hence the bias of $C$, and state whether $C$ over- or underestimates $\mu$.

   [6 marks]

   (b)

   Find the variance of each of $A$, $B$ and $C$, and hence evaluate which estimator you would recommend for $\mu$, justifying your choice.

   [6 marks]

   Total for question 4: 12 marks

End of questions