OCR Further Statistics (Further Statistics) 2020 November

1 The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.

2 A book collector compared the prices of some books, \(\pounds x\), when new in 1972 and the prices of copies of the same books, \(\pounds y\), on a second-hand website in 2018.
The results are shown in Table 1 and are summarised below the table. \begin{table}[h] \end{table} $$n = 12 , \sum x = 9.20 , \sum y = 54.64 , \sum x ^ { 2 } = 8.9950 , \sum y ^ { 2 } = 310.4572 , \sum x y = 46.0545$$

1. It is given that the value of Pearson’s product-moment correlation coefficient for the data is 0.381, correct to 3 significant figures.
   1. State what this information tells you about a scatter diagram illustrating the data.
   2. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018.
2. The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books.

3 Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B . She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219 .

1. Test at the \(10 \%\) significance level whether route A has shorter journey times than route B .
2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid.

4 The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\mathrm { E } ( 3 X ) = 30\) and \(\operatorname { Var } ( 3 X ) = 36\). Determine the value of \(m\) and the value of \(n\). 526 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels.
2. All 26 cards are arranged in a line, in random order.
   1. Show that the probability that all the vowels are next to one another is \(\frac { 1 } { 2990 }\).
   2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other.

6 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\) ? Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
2. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( x \geqslant 8 )\). The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution \(\operatorname { Po } ( 7.2 )\).
3. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
4. State an assumption needed for your answer to part (c) to be valid.
5. Give a reason why the assumption in part (d) may not be valid in practice.

7 A biased spinner has five sides, numbered 1 to 5 . Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(x f\). \begin{table}[h] \end{table}

1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). Elmer carries out a goodness-of-fit test, at the \(5 \%\) level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places. \begin{table}[h]

   \(x\)	1	2	3	4	5	6	\(\geqslant 7\)
   Observed frequency \(O\)	20	15	9	13	10	10	23
   Expected frequency \(E\)	25	18.75	14.063	10.547	7.910	5.933	17.798
   ( \(\mathrm { O } - \mathrm { E } ) ^ { 2 } / \mathrm { E }\)	1	0.75	1.823	0.571	0.552	2.789	1.520

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained.
3. Carry out the test.

8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.