Marine biologists are investigating the correlation between water salinity (x x\,x in parts per thousand, ppt) and the dissolved oxygen concentration (y y\,y in mg/L) in a coastal estuary. The least squares regression line of y y\,y on x x\,x is established as:
y=12.3−0.16xy = 12.3 - 0.16xy=12.3−0.16x
The following summary data from 15 samples are recorded:
∑y=112.5,∑y2=862.0,∑x2=13900,n=15\sum y = 112.5, \quad \sum y^2 = 862.0, \quad \sum x^2 = 13900, \quad n = 15∑y=112.5,∑y2=862.0,∑x2=13900,n=15
Show that Syy=18.25S_{yy} = 18.25Syy=18.25.
Calculate SxxS_{xx}Sxx.
Determine the product moment correlation coefficient (PMCC) between x x\,x and yyy.
A biologist claims that at a salinity level of 10 ppt, the dissolved oxygen concentration will be greater than 10 mg/L.
Explain how the biologist might have reached this conclusion using the regression model.
Another researcher defines the typical range of observed salinity values using the formula:
range=mean±2.5×standard deviation\text{range} = \text{mean} \pm 2.5 \times \text{standard deviation}range=mean±2.5×standard deviation
Using this formula and the given summary statistics, find the minimum and maximum salinity values for this range.
With reference to the range calculated in part (e), comment on the reliability of the biologist’s claim in part (d).
A student suggests using the regression equation y=12.3−0.16xy = 12.3 - 0.16xy=12.3−0.16x to predict the salinity (xxx) of a sample where the oxygen concentration is 6.5 mg/L. Evaluate the student’s suggestion.
Practise Edexcel A Level Maths Regression, Correlation and Hypothesis Testing with exam-style questions for A Level Maths. 100 questions covering Exponential Models, Measuring Correlation, and Hypothesis Testing for Zero Correlation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.