Calculate r from summary statistics

The heights, \(h\) m, of eight children were measured, giving the following values of \(h\): 1.20, 1.12, 1.43, 0.98, 1.31, 1.26, 1.02, 1.41.

1. Find the mean height of the children. [2 marks]
2. Calculate the variance of the heights. [3 marks]

The children were also weighed. It was found that their masses, \(w\) kg, were such that $$\sum w = 324, \quad \sum w^2 = 13532, \quad \sum wh = 403.$$

1. Calculate the product-moment correlation coefficient between \(w\) and \(h\). [4 marks]
2. Comment briefly on the value you have obtained. [1 mark]

The following marks out of 50 were given by two judges to the contestants in a talent contest:

Contestant	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Judge 1 (\(x\))	43	32	40	21	47	11	29	38
Judge 2 (\(y\))	39	25	40	22	36	13	27	32

Given that \(\sum x = 261\), \(\sum x^2 = 9529\) and \(\sum xy = 8373\),

1. calculate the product-moment correlation coefficient between the two judges' marks [5 marks]
2. Find an equation of the regression line of \(x\) on \(y\). [4 marks]

Contestant \(I\) was awarded 45 marks by Judge 2.

1. Estimate the mark that this contestant would have received from Judge 1. [2 marks]
2. Comment, with explanation, on the probable accuracy of your answer. [2 marks]

The heights, \(h\) m, and weights, \(m\) kg, of five men were measured. The results are plotted on the diagram. \includegraphics{figure_3} The results are summarised as follows. \(n = 5\) \(\Sigma h = 9.02\) \(\Sigma m = 377.7\) \(\Sigma h^2 = 16.382\) \(\Sigma m^2 = 28558.67\) \(\Sigma hm = 681.612\)

1. Use the summarised data to calculate the value of the product moment correlation coefficient, \(r\). [3]
2. Comment on your value of \(r\) in relation to the diagram. [2]
3. It was decided to re-calculate the value of \(r\) after converting the heights to feet and the masses to pounds. State what effect, if any, this will have on the value of \(r\). [1]
4. One of the men had height 1.63 m and mass 78.4 kg. The data for this man were removed and the value of \(r\) was re-calculated using the original data for the remaining four men. State in general terms what effect, if any, this will have on the value of \(r\). [1]

The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x\)\%, and the Consumer Price Index (CPI), \(y\)\%, in the United Kingdom from April 2008 to July 2010.

Date	April 2008	July 2008	October 2008	January 2009	April 2009	July 2009	October 2009	January 2010	April 2010	July 2010
UR, \(x\)\%	5.2	5.7	6.1	6.8	7.5	7.8	7.8	7.9	7.8	7.7
CPI, \(y\)\%	3.0	4.4	4.5	3.0	2.3	1.8	1.5	3.5	3.7	3.1

These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x^2 = 503.45 \quad \sum y = 30.8 \quad \sum y^2 = 103.94 \quad \sum xy = 211.9$$

1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(-0.6 < r < -0.5\). [3]
2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement. [1]
3. 1. Calculate the equation of the regression line of \(x\) on \(y\). [3]
   2. Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%. [2]

A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w^2 = 13447 \quad S_{ff} = 42 \quad S_{fw} = 269.5$$

1. Calculate the product moment correlation coefficient between \(f\) and \(w\) [2]
2. Interpret the value of your product moment correlation coefficient. [1]
3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + bf\) [3]
4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² [1]

A set of bivariate data \((X, Y)\) is summarised as follows. \(n = 25\), \(\Sigma x = 9.975\), \(\Sigma y = 11.175\), \(\Sigma x^2 = 5.725\), \(\Sigma y^2 = 46.200\), \(\Sigma xy = 11.575\)

1. Calculate the value of Pearson's product-moment correlation coefficient. [1]
2. Calculate the equation of the regression line of \(y\) on \(x\). [2]

It is desired to know whether the regression line of \(y\) on \(x\) will provide a reliable estimate of \(y\) when \(x = 0.75\).

1. State one reason for believing that the estimate will be reliable. [1]
2. State what further information is needed in order to determine whether the estimate is reliable. [1]