Linear combinations and the sample mean Problem set B Worked solutions

The random variable \(X\) has \(P(X=x)=\dfrac{x}{15}\) for \(x=1, 2, 3, 4, 5\).

1. (a) Find \(E(X)\).

   • \(E(X)=\sum xP(X=x)=1(\tfrac{1}{15})+2(\tfrac{2}{15})+3(\tfrac{1}{5})+4(\tfrac{4}{15})+5(\tfrac{1}{3})=\tfrac{11}{3}\)

2. (b) The random variable \(Y\) is given by \(Y=3X+8\). Find the mean of \(Y\).

   • \(E(Y)=3(\tfrac{11}{3})+8=19\)

3. (c) Given that \(\mathrm{Var}(X)=\tfrac{14}{9}\), find the variance of \(Y\).

   • \(\mathrm{Var}(Y)=3^2(\tfrac{14}{9})=14\)

\(X\) and \(Y\) are independent. \(E(X)=10\), \(\mathrm{Var}(X)=2.5\), \(E(Y)=4\) and \(\mathrm{Var}(Y)=0.6\).

1. (a) Find \(E(Z)\) where \(Z=4X-Y-3\).

   • \(E(Z)=4(10)-4-3=33\)

2. (b) Find \(\mathrm{Var}(Z)\).

   • \(\mathrm{Var}(Z)=4^2(2.5)+0.6=40.6\)

A rower's time for 500 m has mean 98 s and standard deviation 4.5 s. The mean of 9 independent times is calculated. Find the mean and standard deviation of this sample mean.

• \(E(\bar{X})=98\)
• \(\mathrm{Var}(\bar{X})=\dfrac{4.5^2}{9}=2.25\)
• \(\text{sd}=\sqrt{2.25}=1.5\)

The random variable \(M\) has mean 6.5 and variance 1.2. The random variable \(N\) is such that \(M+4N=40\). Find the mean and variance of \(N\).

• \(N=\dfrac{40-M}{4}=10-0.25M\)
• \(E(N)=10-0.25(6.5)=8.375\)
• \(\mathrm{Var}(N)=(0.25)^2(1.2)=0.075\)

The mass of a pebble on a beach has mean 35 g and standard deviation 4 g. Two pebbles are chosen at random. Find the mean and standard deviation of the mass of the first pebble minus the mass of the second.

• \(E(X_1-X_2)=35-35=0\)
• \(\mathrm{Var}(X_1-X_2)=4^2+4^2=32\)
• \(\text{sd}=\sqrt{32}=5.66\)

Three runners take part in a relay. The first runner's time has mean 12.4 s and variance 0.36 s². The second has mean 11.8 s and variance 0.25 s². The third has mean 12.9 s and variance 0.49 s². The times are independent. Find the mean and standard deviation of the total relay time.

• \(E(T)=12.4+11.8+12.9=37.1\)
• \(\mathrm{Var}(T)=0.36+0.25+0.49=1.1\)
• \(\text{sd}=\sqrt{1.1}=1.05\)

Bottles of juice are packed in crates of 18. The crate itself has mass exactly 650 g. The mass of one bottle has mean 410 g and standard deviation 6 g. Find the mean and standard deviation of the mass of a full crate.

• \(E(T)=18(410)+650=8030\)
• \(\mathrm{Var}(T)=18(36)=648\)
• \(\text{sd}=\sqrt{648}=25.5\)

A trolley with mass exactly 24 kg carries 6 small boxes and 4 large boxes. A small box has mean mass 3.5 kg and standard deviation 0.4 kg. A large box has mean mass 11 kg and standard deviation 1.2 kg. All masses are independent. Find the mean and standard deviation of the total mass of the loaded trolley.

• \(E(T)=24+6(3.5)+4(11)=89\)
• \(\mathrm{Var}(T)=0+6(0.16)+4(1.44)=6.72\)
• \(\text{sd}=\sqrt{6.72}=2.59\)

Pip cycles to work with a mean time of 17 minutes and standard deviation 2.5 minutes. Quin drives with a mean time of 24 minutes and standard deviation 3 minutes. Each adds up the journey times over a four-day working week. Assume all journey times are independent. Find the mean and standard deviation of how much less time Pip spends travelling than Quin over the week.

• \(D=\text{Quin's total}-\text{Pip's total}\)
• \(E(D)=4(24)-4(17)=28\)
• \(\mathrm{Var}(D)=4(3^2)+4(2.5^2)=61\)
• \(\text{sd}=\sqrt{61}=7.81\)

A spinner has mean score 3.5 and standard deviation 0.6. Ines spins once and multiplies the score by 5. Jo spins three times and adds the scores. Find the mean and standard deviation of Ines's result minus Jo's result.

• \(D=5X-(X_1+\dots+X_3)\)
• \(E(D)=5(3.5)-3(3.5)=7\)
• \(\mathrm{Var}(D)=5^2(0.6^2)+3(0.6^2)=10.08\)
• \(\text{sd}=\sqrt{10.08}=3.17\)

The standard deviation of the mean length of a sample of 9 screws is 8 mm less than the standard deviation of the length of a single screw. Find the standard deviation of the length of a single screw.

• \(\sigma-\dfrac{\sigma}{\sqrt{9}}=8\)
• \(\sigma\left(1-\tfrac{1}{3}\right)=8\)
• \(\sigma=12\)

The length of a certain component has variance 4.2. A technician wants the standard deviation of the mean length of a sample to be less than 0.15. Find the minimum sample size.

• \(\sqrt{\dfrac{4.2}{n}}<0.15\)
• \(n>\dfrac{4.2}{0.0225}=186.67\)
• \(n=187\)

A sample of size \(n\) is taken. When the sample size is increased by 90, the standard deviation of the sample mean becomes a quarter of its original value. Find \(n\).

• \(\dfrac{\sigma}{\sqrt{n+90}}=\dfrac{1}{4}\cdot\dfrac{\sigma}{\sqrt{n}}\)
• \(\sqrt{n+90}=4\sqrt{n}\)
• \(n+90=16n,\ \ 15n=90,\ \ n=6\)