Estimators, bias and the sampling distribution of the meanEdexcel International A Level Further Maths: Mind map

Exam questions on Estimators, bias and the sampling distribution of the mean

1. A random sample of 8 observations of a variable $X$ gives $\sum x=100$ and $\sum x^2=1292$.
   Calculate an estimate of the standard error of the sample mean.2 marks
2. $X_1$, $X_2$ and $X_3$ are independent observations from a population with mean $\mu$ and variance $\sigma^2$.
   Show that $T=\frac{X_1+X_2+X_3}{2}$ is a biased estimator of $\mu$, and state the bias.2 marks
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.
   State the distribution of $\bar X$, and find $P(\bar X>156)$.3 marks