Questions — OCR Further Statistics (108 questions)

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OCR Further Statistics 2017 Specimen Q2
6 marks Standard +0.3
The mass \(J\) kg of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass \(K\) kg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.
  1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg. [3]
  2. Find the probability that the mass of one bag of King Edward potatoes is more than 75\% of the mass of one bag of Jersey potatoes. [3]
OCR Further Statistics 2017 Specimen Q3
8 marks Standard +0.3
A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in £, that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.
  1. Find the probability distribution of the amount of money won by the contestant. [3]
  2. The contestant pays £5 for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice. [5]
OCR Further Statistics 2017 Specimen Q4
7 marks Challenging +1.2
A psychologist investigated the scores of pairs of twins on an aptitude test. Seven pairs of twins were chosen randomly, and the scores are given in the following table.
Elder twin65376079394088
Younger twin58396162502684
  1. Carry out an appropriate Wilcoxon test at the 10\% significance level to investigate whether there is evidence of a difference in test scores between the elder and the younger of a pair of twins. [6]
  2. Explain the advantage in this case of a Wilcoxon test over a sign test. [1]
OCR Further Statistics 2017 Specimen Q5
8 marks Standard +0.3
The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).
  1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. [2]
Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).
    1. Write down an expression for P\((X = r)\). [1]
    2. Hence find P\((X = 3)\). [1]
  2. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match. [4]
OCR Further Statistics 2017 Specimen Q6
7 marks Standard +0.3
A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by \(X\).
  1. In the case when \(w = 2\),
    1. write down the distribution of \(X\), [1]
    2. find \(P(3 < X \leq 7)\). [2]
  2. In the case when E\((X) = 2\), determine the value of \(w\). [2]
  3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour. [2]
OCR Further Statistics 2017 Specimen Q7
9 marks Standard +0.3
Sweet pea plants grown using a standard plant food have a mean height of 1.6 m. A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$n = 49$$ $$\sum x = 74.48$$ $$\sum x^2 = 120.8896$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m. [9]
OCR Further Statistics 2017 Specimen Q8
15 marks Standard +0.8
A continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} 0.8e^{-0.8x} & x \geq 0, \\ 0 & x < 0. \end{cases}$$
  1. Find the mean and variance of \(X\). [4]
The lifetime of a certain organism is thought to have the same distribution as \(X\). The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.
Range\(0 \leq x < 1\)\(1 \leq x < 2\)\(2 \leq x < 3\)\(3 \leq x < 4\)\(x \geq 4\)
Observed24221031
Expected33.04014.8466.6712.9972.446
  1. Show how the expected frequency for \(1 \leq x < 2\) is obtained. [4]
  2. Carry out a goodness of fit test at the 5\% significance level. [7]
OCR Further Statistics 2017 Specimen Q9
9 marks Challenging +1.2
The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x < 0, \\ \frac{1}{16}x^2 & 0 \leq x \leq 4, \\ 1 & x > 4. \end{cases}$$
  1. The random variable \(Y\) is defined by \(Y = \frac{1}{X^2}\). Find the cumulative distribution function of \(Y\). [5]
  2. Show that E\((Y)\) is not defined. [4]