OCR S1 (Statistics 1)

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p\). 16 shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   1. at least 8,
   2. between 4 and 9 inclusive.
   3. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 . \section*{June 2005}

4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities. $$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

1. Calculate the product moment correlation coefficient.
2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54. State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate. \section*{June 2005}

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-04_1335_1319_404_413} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)? \section*{June 2005}

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-05_858_980_573_585}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

7 A committee of 7 people is to be chosen at random from 18 volunteers.

1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
3. include exactly 5 people from Worcester,
4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
5. Jenny estimates that her chances of success are as follows.
   • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
   • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
   • John estimates that his chances of success are as follows.
   • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
   • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
   Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
7. Describe briefly how the heights of the second group of plants compare with the first.
8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
10. The probability distribution of \(X\) is given in the table below.

    \(x\)	2	3	4	5
    \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)

    Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

8 The table shows the population, \(x\) million, of each of nine countries in Western Europe together with the population, \(y\) million, of its capital city. $$\left[ n = 9 , \Sigma x = 337.5 , \Sigma x ^ { 2 } = 18959.11 , \Sigma y = 28.3 , \Sigma y ^ { 2 } = 161.65 , \Sigma x y = 1533.76 . \right]$$

1. (a) Calculate Spearman's rank correlation coefficient, \(r _ { s }\).
   (b) Explain what your answer indicates about the populations of these countries and their capital cities.
2. Calculate the product moment correlation coefficient, \(r\). The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-09_936_881_1162_632}
3. By considering the diagram, state the effect on the value of the product moment correlation coefficient, \(r\), if the data for France and the United Kingdom were removed from the calculation.
4. In a certain country in Africa, most people live in remote areas and hence the population of the country is unknown. However, the population of the capital city is known to be approximately 1 million. An official suggests that the population of this country could be estimated by using a regression line drawn on the above scatter diagram.
   (a) State, with a reason, whether the regression line of \(y\) on \(x\) or the regression line of \(x\) on \(y\) would need to be used.
   (b) Comment on the reliability of such an estimate in this situation. 1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$
5. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
6. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
7. Find the values of \(\bar { x }\) and \(\bar { y }\). 2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that
8. the second disc is black, given that the first disc was black,
9. the second disc is black,
10. the two discs are of different colours. 3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
11. How many different arrangements of the letters are possible?
12. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
13. Find the probability that at least one of these 2 cards has D printed on it. 4
14. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
15. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
16. The random variable \(Z\) has the distribution \(B ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\). 5 The probability distribution of a discrete random variable, \(X\), is given in the table.

    \(x\)	0	1	2	3
    \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(q\)

    It is given that the expectation, \(\mathrm { E } ( X )\), is \(1 \frac { 1 } { 4 }\).
17. Calculate the values of \(p\) and \(q\).
18. Calculate the standard deviation of \(X\).