Estimators, bias and the sampling distribution of the meanEdexcel International A Level Further Maths: Subtopic test

1. 1
   A random sample of 8 observations of a variable $X$ gives $\sum x=100$ and $\sum x^2=1292$.

   (a)

   What is an unbiased estimate of the population mean?

   [1 mark]

   • A$161.5$
   • B$1292$
   • C$100$
   • D$12.5$

   (b)

   What is an unbiased estimate of the population variance?

   [1 mark]

   • A$6$
   • B$5.25$
   • C$42$
   • D$2.45$

   (c)

   Calculate an estimate of the standard error of the sample mean.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $X_1$, $X_2$ and $X_3$ are independent observations from a population with mean $\mu$ and variance $\sigma^2$.

   (a)

   Which of the following is an unbiased estimator of $\mu$?

   [1 mark]

   • A$\frac{X_1+X_2+X_3}{2}$
   • B$\frac{X_1+2X_2}{3}$
   • C$\frac{X_1+X_2}{3}$
   • D$\frac{X_1+X_2+X_3}{4}$

   (b)

   What is the standard error of the sample mean $\bar X=\frac{X_1+X_2+X_3}{3}$?

   [1 mark]

   • A$\frac{\sigma}{3}$
   • B$\frac{\sigma^2}{3}$
   • C$\frac{\sigma}{\sqrt3}$
   • D$\sigma\sqrt3$

   (c)

   Show that $T=\frac{X_1+X_2+X_3}{2}$ is a biased estimator of $\mu$, and state the bias.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The masses of apples from an orchard are Normally distributed with mean $150$ g and standard deviation $20$ g. Random samples of $16$ apples are taken, and the sample mean $\bar X$ is calculated for each sample.

   (a)

   State the distribution of $\bar X$, and find $P(\bar X>156)$.

   [3 marks]

   (b)

   (i) Find the probability that the sample mean of 16 apples is within $4$ g of $150$ g.
   (ii) Find the probability that a single apple has a mass within $4$ g of $150$ g, and explain what a comparison of the two probabilities shows about the sample mean.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The times, in seconds, taken by 10 randomly chosen runners to complete a sprint are summarised by $\sum x=415$ and $\sum x^2=17\,305$. The times are modelled as a random sample from a population with unknown mean $\mu$ and unknown variance $\sigma^2$.

   (a)

   (i) Calculate unbiased estimates of $\mu$ and $\sigma^2$.
   (ii) Calculate an estimate of the standard error of the sample mean.

   [6 marks]

   (b)

   (i) Calculate the value of $\frac{1}{n}\sum(x-\bar x)^2$ for this sample.
   (ii) Explain why this is not used as an estimator of $\sigma^2$.
   (iii) Assuming $\sigma$ is unchanged, find the sample size needed to halve the standard error of the sample mean.

   [6 marks]

   Total for question 4: 12 marks

End of questions