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.
| Germany | United Kingdom | France | Italy | Spain | The Netherlands | Portugal | Austria | Switzerland |
| \(x\) | 82.1 | 59.2 | 59.1 | 56.7 | 39.2 | 15.9 | 9.9 | 8.1 | 7.3 |
| \(y\) | 3.5 | 7.0 | 9.0 | 2.7 | 2.9 | 0.8 | 0.7 | 1.6 | 0.1 |
$$\left[ n = 9 , \Sigma x = 337.5 , \Sigma x ^ { 2 } = 18959.11 , \Sigma y = 28.3 , \Sigma y ^ { 2 } = 161.65 , \Sigma x y = 1533.76 . \right]$$
- (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. - 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}
- 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.
- 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.
$$\begin{array} { c c }
y \text { on } x : & y = - 0.6 x + 13.0 \\
x \text { on } y : & x = - 1.6 y + 21.0
\end{array}$$
- State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
- Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
- 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
- the second disc is black, given that the first disc was black,
- the second disc is black,
- 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.
- How many different arrangements of the letters are possible?
- 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.
- Find the probability that at least one of these 2 cards has D printed on it.
4
- The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
- The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
- 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 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 4 }\) | \(p\) | \(q\) |
- Calculate the values of \(p\) and \(q\).
- Calculate the standard deviation of \(X\).