A marine biologist is investigating the relationship between water depth and the concentration of a specific dissolved mineral in a coastal lake. The concentration, c c\,c mg/L, was measured at various depths, d d\,d metres, below the lake surface. The results for a random sample of 8 measurements are summarised in the table below.
| Depth (d d\,d m) | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
|---|---|---|---|---|---|---|---|---|
| Concentration (c c\,c mg/L) | 15.2 | 14.1 | 12.8 | 11.5 | 10.2 | 8.8 | 7.5 | 6.3 |
[You may use: ∑d=360\sum d = 360∑d=360, ∑c=86.4\sum c = 86.4∑c=86.4, ∑d2=20400\sum d^2 = 20400∑d2=20400, ∑dc=3345\sum dc = 3345∑dc=3345]
Calculate SddS_{dd}Sdd and SdcS_{dc}Sdc.
Find the equation of the least squares regression line of c c\,c on d d\,d in the form c=a+bdc = a + bdc=a+bd.
Give a practical interpretation of the gradient of your regression line in the context of this study.
A technician takes a measurement at a depth of 45 metres. Predict the mineral concentration at this depth.
A researcher suggests that at a depth of 150 metres, the mineral concentration will have decreased to less than 2.0 mg/L. Comment on the validity of this claim.
