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IB Mathematics: Applications and Interpretation HL · AHL 8: Statistics 2 · AHL 8.6

Name Date

Linear combinations and the sample mean Problem set A

Define each random variable, apply the mean and variance rules, and take a square root only at the end for a standard deviation.

Difficulty: first skills applying them stretch

Skills

  1. The random variable \(X\) has \(P(X=x)=\dfrac{5-x}{10}\) for \(x=1, 2, 3, 4\).

    1. (a) Find \(E(X)\).
    2. (b) The random variable \(Y\) is given by \(Y=6X-1\). Find the mean of \(Y\).
    3. (c) Given that \(\mathrm{Var}(X)=1\), find the variance of \(Y\).

  2. \(X\) and \(Y\) are independent. \(E(X)=3.5\), \(\mathrm{Var}(X)=1.2\), \(E(Y)=-2\) and \(\mathrm{Var}(Y)=0.8\).

    1. (a) Find \(E(Z)\) where \(Z=3X-2Y+1\).
    2. (b) Find \(\mathrm{Var}(Z)\).

  3. A machine fills tubs of margarine so that the mean mass of a tub is 250 g and the standard deviation is 6.4 g. A sample of 16 tubs is taken and the mean mass is calculated. Find the mean and standard deviation of the sample mean.

  4. The random variable \(R\) has mean 14 and variance 3.6. The random variable \(S\) is such that \(3R+2S=60\). Find the mean and variance of \(S\).

  5. At a large school, the reaction time of a student has mean 0.24 s and standard deviation 0.05 s. Two students are chosen at random. Find the mean and standard deviation of the first student's time minus the second student's time.

In context

  1. A garden path is laid in three layers: sand, mortar and a paving slab. The thickness of the sand has mean 4.2 cm and variance 0.09 cm². The mortar has mean 1.5 cm and variance 0.04 cm². The slab has mean 3.8 cm and variance 0.16 cm². The thicknesses are independent. Find the mean and standard deviation of the total thickness of the path.

  2. Yoghurt pots are packed in trays of 24. The tray itself has mass exactly 85 g. The mass of one pot has mean 126 g and standard deviation 2.5 g. Find the mean and standard deviation of the mass of a full tray.

  3. A minibus with an empty mass of exactly 1850 kg carries 7 children and 3 adults. A child has mean mass 38 kg and standard deviation 5 kg. An adult has mean mass 71 kg and standard deviation 9 kg. All masses are independent. Find the mean and standard deviation of the total mass of the loaded minibus.

  4. Mia's bus journey to school has mean 26 minutes and standard deviation 4 minutes. Noor's tram journey has mean 33 minutes and standard deviation 5 minutes. Each adds up her journey times over a six-day week. Assume all journey times are independent. Find the mean and standard deviation of how much less time Mia spends travelling than Noor over the week.

Stretch

  1. A weighted die has mean score 3.2 and standard deviation 0.9. Chloe rolls the die once and multiplies the score by 4. Dev rolls the die four times and adds the scores. Find the mean and standard deviation of Chloe's result minus Dev's result.

  2. The standard deviation of the mean mass of a sample of 4 pencils is 3 g less than the standard deviation of the mass of a single pencil. Find the standard deviation of the mass of a single pencil.

  3. A test score has variance 2.5. A teacher wants the standard deviation of the mean score of a sample to be less than 0.2. Find the minimum sample size.

  4. A sample of size \(n\) is taken. When the sample size is increased by 32, the standard deviation of the sample mean becomes a third of its original value. Find \(n\).

jmaths.xyz · AI HL · AHL 8.6 Mixed practice: linear combinations and the sample mean