For an ecological study, Ernesto measured the average concentration \((y)\) of the fine dust, \({\text{PM}}10\), in the air at different distances \((x)\) from a power plant. His data are represented on the following scatter diagram. The concentration of \({\text{PM}}10\) is measured in micrograms per cubic metre and the distance is measured in kilometres.

His data are also listed in the following table.

Use the scatter diagram to find the value of \(a\) and of \(b\) in the table.

Calculate

i)      \({\bar x}\) , the mean distance from the power plant;

ii)     \({\bar y}\) , the mean concentration of \({\text{PM}}10\) ;

iii)    \(r\) , the Pearson’s product–moment correlation coefficient.

Write down the equation of the regression line \(y\) on \(x\) .

Ernesto’s school is located \(14\,{\text{km}}\) from the power plant. He uses the equation of the regression line to estimate the concentration of \({\text{PM}}10\) in the air at his school.

i)     Calculate the value of Ernesto’s estimate.

ii)    State whether Ernesto’s estimate is reliable. Justify your answer.

\(a = 4.2\,;\,\,b = 74\)             (A1)(A1)

i)      \(5.91\,({\text{km}})\)       (A1)(ft)

ii)     \(88\) (micrograms per cubic metre)      (A1)(ft)

Note: Follow through from part (a) irrespective of working seen.

iii)    \( - 0.956\,\,\,\,( - 0.955528...)\)        (G2)(ft)

Note: Follow through from part (a) irrespective of working seen.

\(y =  - 5.39x + 120\,\,\,\,(y =  - 5.38955...x + 119.852...)\)           (A1)(ft)(A1)(ft)

Note: Award (A1)(ft) for \( - 5.39\). Award (A1)(ft) for \(120\). If answer is not an equation award at most (A1)(ft)(A0). Follow through from part (a) irrespective of working seen.

i)     \( - 5.38955... \times 14 + 119.852...\)        (M1)

Note: Award (M1) for correct substitution into their regression line.

\( = 44.4\,\,(44.3984...)\)          (A1)(ft)(G2)

Note: Follow through from part (c). Accept \(44.5\,\,(44.54)\) from use of \(3\) significant figure values.

ii)     Ernesto’s estimate is not reliable        (A1)

this is extrapolation        (R1)

OR

\(14\,{\text{km}}\) is not within the range (outside the domain) of distances given       (R1)

Note: Do not accept “\(14\) is too high” or “\(14\) is an outlier” or “result not valid/not reliable” if explanation not given. Do not award (A1)(R0). Do not accept reasoning based on the strength of \(r\).

Question 1: Reading scatter diagram, mean, correlation and regression line.
The majority of the candidates scored very well on this question. There were only a few candidates who read the diagram incorrectly. The most common mistake in parts (b), (c) and (d)(i) were rounding errors, sometimes resulting in candidates losing follow-through marks when working was not presented. Part (d)(ii) was answered incorrectly by most candidates. The most common incorrect answer was based on strong correlation. Some commented on the trend of decreasing PM10 values for increasing distances, showing lack of understanding about extrapolation.

Question 1: Reading scatter diagram, mean, correlation and regression line.
The majority of the candidates scored very well on this question. There were only a few candidates who read the diagram incorrectly. The most common mistake in parts (b), (c) and (d)(i) were rounding errors, sometimes resulting in candidates losing follow-through marks when working was not presented. Part (d)(ii) was answered incorrectly by most candidates. The most common incorrect answer was based on strong correlation. Some commented on the trend of decreasing PM10 values for increasing distances, showing lack of understanding about extrapolation.

Question 1: Reading scatter diagram, mean, correlation and regression line.
The majority of the candidates scored very well on this question. There were only a few candidates who read the diagram incorrectly. The most common mistake in parts (b), (c) and (d)(i) were rounding errors, sometimes resulting in candidates losing follow-through marks when working was not presented. Part (d)(ii) was answered incorrectly by most candidates. The most common incorrect answer was based on strong correlation. Some commented on the trend of decreasing PM10 values for increasing distances, showing lack of understanding about extrapolation.

Question 1: Reading scatter diagram, mean, correlation and regression line.
The majority of the candidates scored very well on this question. There were only a few candidates who read the diagram incorrectly. The most common mistake in parts (b), (c) and (d)(i) were rounding errors, sometimes resulting in candidates losing follow-through marks when working was not presented. Part (d)(ii) was answered incorrectly by most candidates. The most common incorrect answer was based on strong correlation. Some commented on the trend of decreasing PM10 values for increasing distances, showing lack of understanding about extrapolation.