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=12.3−0.16x y = 12.3 - 0.16x y=12.3−0.16xThe 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=15Show 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 deviationUsing 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.
