The sample mean and its standard error Problem set A

A random variable \(X\) has the mean and variance given in each part. Find \(E(\bar{X})\) and \(\mathrm{Var}(\bar{X})\) for a sample of size \(n\).

1. (a) \(E(X)=7.5,\ \mathrm{Var}(X)=2.4,\ n=8\)
2. (b) \(E(X)=-12.6,\ \mathrm{Var}(X)=0.9,\ n=20\)

Find \(E(\bar{X})\) and \(\mathrm{Var}(\bar{X})\) for the mean of a sample of size \(n\) from

1. (a) \(X\sim N(20,\,16),\ n=8\)
2. (b) \(X\sim N(5.5,\,0.36),\ n=15\)

Find \(E(X)\) and \(\mathrm{Var}(X)\). Then find \(E(\bar{X})\) and \(\mathrm{Var}(\bar{X})\) for the mean of a sample of size \(n\) of observations of \(X\).

1. (a) \(X\sim B(10,\,0.4),\ n=12\)
2. (b) \(X\sim B(25,\,0.2),\ n=10\)

The table shows the probability distribution of the random variable \(X\).

1. (a) Find \(E(X)\).
2. (b) The random variable \(Y\) is given by \(Y=5X-2\). Find \(E(Y)\).
3. (c) Given that \(\mathrm{Var}(X)=0.84\), find \(\mathrm{Var}(Y)\).

The random variable \(W\) has \(P(W=w)=\dfrac{k}{w}\) for \(w=1, 2, 3, 6\).

1. (a) Find the value of \(k\).
2. (b) Find \(E(W)\).
3. (c) Given that \(\mathrm{Var}(W)=2\), find the variance of \(V=6W-5\).

The mass of a bag of rice has mean 1.02 kg and standard deviation 0.04 kg. An inspector weighs a sample of 25 bags and finds their mean mass \(\bar{X}\).

1. (a) Find \(E(\bar{X})\) and \(\mathrm{Var}(\bar{X})\).
2. (b) Find the standard error of the mean.
3. (c) The inspector takes a sample of 100 bags instead. Find the new standard error and say how it compares with part (b).

A factory makes 50 bolts each day. Each bolt is faulty with probability 0.08, independently. Let \(X\) be the number of faulty bolts in one day. The mean number of faulty bolts per day is found over a sample of 30 days.

1. (a) Find \(E(X)\) and \(\mathrm{Var}(X)\).
2. (b) Find the expected value and variance of the mean number of faulty bolts per day over the 30 days.

The time a worker takes to travel to work is modelled by \(X\sim N(32,\,6^2)\), in minutes. The mean time \(\bar{X}\) over 20 journeys is recorded.

1. (a) Find \(E(\bar{X})\) and \(\mathrm{Var}(\bar{X})\).
2. (b) Find the standard error of \(\bar{X}\).

A mechanic charges £45 per hour plus a fixed call-out fee of £30. The time \(X\) hours of a job has mean 2.4 and variance 0.64. The cost of a job is \(C=45X+30\). The mean cost \(\bar{C}\) of a sample of 16 jobs is found. Find \(E(\bar{C})\), \(\mathrm{Var}(\bar{C})\) and the standard error of \(\bar{C}\).