OCR Further Statistics (Further Statistics) 2017 Specimen

2 The mass \(J \mathrm {~kg}\) 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 Kkg 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 .
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.

5 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. Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
2. (a) Write down an expression for \(\mathrm { P } ( X = r )\).
   (b) Hence find \(\mathrm { P } ( X = 3 )\).
3. 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.

6 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 \(w = 2\),
   (a) write down the distribution of \(X\),
   (b) find \(P ( 3 < X \leq 7 )\).
2. In the case when \(\mathrm { E } ( X ) = 2\), determine the value of \(w\).
3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour.

7 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. $$\begin{aligned} n & = 49
\Sigma x & = 74.48
\Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$ 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 .