Calculate regression line equation

6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. The table shows the values of the efficiency rating, y , after the machine has carried out its task various numbers of times, \(x\) $$\left[ n = 8 , \Sigma x = 60 , \Sigma y = 72 , \Sigma x ^ { 2 } = 1148 , \Sigma y ^ { 2 } = 810 , \Sigma x y = 767 . \right]$$ These data are illustrated in the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-4_769_1328_760_411}

1. (a) Calculate the value of r , the product moment correlation coefficient.
   (b) Without calculation, state with a reason the value of \(\mathrm { r } _ { \mathrm { s } ^ { \prime } }\) Spearman's rank correlation coefficient.
2. A researcher suggests that the data for \(\mathrm { x } = 0\) and \(\mathrm { x } = 1\) should be ignored. Without cal culation, state with a reason what effect this would have on the value of
   (a) \(r\),
   (b) \(r _ { s }\).
3. Use the diagram to estimate the value of y when \(\mathrm { x } = 29\).
4. Jack finds the equation of the regression line of y on xf for all the data, and uses it to estimate the value of \(y\) when \(x = 29\). Without calculation, state with a reason whether this estimate or the one found in part (iii) will be the more reliable.

1. The table shows the price of a bottle of milk, \(m\) pence, and the price of a loaf of bread, \(b\) pence, for 8 different years.

(You may use \(\mathrm { S } _ { b b } = 3083.875\) and \(\mathrm { S } _ { m m } = 305.5\) )

1. Find the exact value of \(\sum b m\)
2. Find \(\mathrm { S } _ { b m }\)
3. Calculate the product moment correlation coefficient between \(b\) and \(m\)
4. Interpret the value of the correlation coefficient. A ninth year is added to the data set. In this year the price of the bottle of milk is 46 pence and the price of a loaf of bread is 175 pence.
5. Without further calculation, state whether the value of the product moment correlation coefficient will increase, decrease or stay the same when all nine years are used. Give a reason for your answer.

1. The average minimum monthly temperature, \(x\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), and the average maximum monthly temperature, \(y\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), in Kolkata were recorded for 12 months.

Some of the summary statistics are given below. $$\sum x = 862 \quad \sum x ^ { 2 } = 62802 \quad \mathrm {~S} _ { y y } = 413.67 \quad S _ { x y } = 512.67 \quad n = 12$$

1. 1. Calculate the mean of the 12 values of the average minimum
      monthly temperature.
   2. Show that the standard deviation of the 12 values of the average minimum monthly temperature is \(8.57 ^ { \circ } \mathrm { F }\) to 3 significant figures.
2. Calculate the product moment correlation coefficient between \(x\) and \(y\) For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ) to degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ). The formula used was $$c = \frac { 5 } { 9 } ( f - 32 )$$ where \(f\) is the temperature in \({ } ^ { \circ } \mathrm { F }\) and \(c\) is the temperature in \({ } ^ { \circ } \mathrm { C }\)
3. Use this formula and the values from part (a) to calculate, in \({ } ^ { \circ } \mathrm { C }\), the mean and the standard deviation of the 12 values of the average minimum monthly temperature in Kolkata.
   Give your answers to 3 significant figures. Given that
   • \(u\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(x\)
   • \(\quad v\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(y\)
   • state, giving a reason, the product moment correlation coefficient between \(u\) and \(v\)

5 Tariq collected information about typical prices, \(\pounds y\) million, of four-bedroomed houses at varying distances, \(x\) miles, from a large city. He chose houses at 10 -mile intervals from the city. His results are shown below. $$n = 8 \quad \Sigma x = 360 \quad \Sigma x ^ { 2 } = 20400 \quad \Sigma y = 6.8 \quad \Sigma y ^ { 2 } = 6.88 \quad \Sigma x y = 241$$

1. Use an appropriate formula to calculate the product moment correlation coefficient, \(r\), showing that \(- 1.0 < r < - 0.9\).
2. State what this value of \(r\) shows in this context.
3. Tariq decides to recalculate the value of \(r\) with the house prices measured in hundreds of thousands of pounds, instead of millions of pounds. State what effect, if any, this will have on the value of \(r\).
4. Calculate the equation of the regression line of \(y\) on \(x\).
5. Explain why the regression line of \(y\) on \(x\), rather than \(x\) on \(y\), should be used for estimating a value of \(x\) from a given value of \(y\).

1. As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where \(t\) represents time and \(\pounds m\) the amount spent over \(\pounds 20\).

1. Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.
2. Calculate \(S _ { t t } , S _ { m m }\) and \(S _ { t m }\). $$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
3. Calculate the value of the product moment correlation coefficient between \(t\) and \(m\).
4. Write down the value of the product moment correlation coefficient between \(t\) and the actual amount spent. Give a reason to justify your value. On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178
5. Give an interpretation to both of these coefficients.
6. Suggest a practical reason why these two values are so different.

4 Ashok is a work-experience student with an organisation that offers two separate professional examination papers, I and II. For each of a random sample of 12 students, A to L , he records the mark, \(x\) per cent, achieved on Paper I, and the mark, \(y\) per cent, achieved on Paper II.

1. 1. Calculate the value of the product moment correlation coefficient, \(r\), between \(x\) and \(y\).
   2. Interpret your value of \(r\) in the context of this question.
2. 1. Give two possible advantages of plotting data on a graph before calculating the value of a product moment correlation coefficient.
   2. Complete the plotting of Ashok's data on the scatter diagram on page 5.
   3. State what is now revealed by the scatter diagram.
3. Ashok subsequently discovers that students A to F have a more scientific background than students G to L. With reference to your scatter diagram, estimate the value of the product moment correlation coefficient for each of the two groups of students. You are not expected to calculate the two values.

   \cline { 2 - 7 } \multicolumn{1}{c|}{}	\(\mathbf { G }\)	\(\mathbf { H }\)	\(\mathbf { I }\)	\(\mathbf { J }\)	\(\mathbf { K }\)	\(\mathbf { L }\)
   \(\boldsymbol { x }\)	60	54	70	71	82	85
   \(\boldsymbol { y }\)	49	60	54	44	49	36

   \section*{Examination Marks}
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