Drilling down beneath a lake in Alaska yields chemical evidence of past changes

Question

Drilling down beneath a lake in Alaska yields chemical evidence of past changes in climate. Biological silicon, left by the skeletons of single-celled creatures called diatoms, measures the abundance of life in the lake. A rather complex variable based on the ratio of certain isotopes relative to ocean water gives an indirect measure of moisture, mostly from snow. As we drill down, we look farther into the past. Here are data from 2300 to 12,000 years ago:

(b) Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)


What is the correlation without this point? (Round your answer to two decimal places.)

Isotope
(%) Silicon
(mg/g) Isotope
(%) Silicon
(mg/g) Isotope
(%) Silicon
(mg/g) 19.90 99 20.71 152 21.63 222 19.84 102 20.80 263 21.63 233 19.46 116 20.86 271 21.19 188 20.20 141 21.28 296 19.37 335

Explanation / Answer

b) The correlation with all the points is -0.3383981 or -0.34.

> cor(x)
Isotope Silicon
Isotope 1.0000000 -0.3383981
Silicon -0.3383981 1.0000000

Now, the correlation of the data by removing the 1,2,....12 th observation we get the correlation to be as follows:

1. Isotope Silicon
Isotope 1.0000000 -0.2629632
Silicon -0.2629632 1.0000000
2. Isotope Silicon
Isotope 1.0000000 -0.2564478
Silicon -0.2564478 1.0000000
3. Isotope Silicon
Isotope 1.0000000 -0.2263859
Silicon -0.2263859 1.0000000
4. Isotope Silicon
Isotope 1.000000 -0.316018
Silicon -0.316018 1.000000
5. Isotope Silicon
Isotope 1.0000000 -0.3561828
Silicon -0.3561828 1.0000000
6. Isotope Silicon
Isotope 1.0000000 -0.3278102
Silicon -0.3278102 1.0000000
7. Isotope Silicon
Isotope 1.0000000 -0.3217999
Silicon -0.3217999 1.0000000
8. Isotope Silicon
Isotope 1.0000000 -0.2638554
Silicon -0.2638554 1.0000000
9. Isotope Silicon
Isotope 1.0000000 -0.3363229
Silicon -0.3363229 1.0000000
10. Isotope Silicon
Isotope 1.0000000 -0.3179746
Silicon -0.3179746 1.0000000
11. Isotope Silicon
Isotope 1.0000000 -0.3625423
Silicon -0.3625423 1.0000000
12. Isotope Silicon
Isotope 1.0000000 -0.7800649
Silicon -0.7800649 1.0000000

Thus , clearly the outlier in the data is the 12th observation namely:

Isotope Silicon
12 -19.37 335

The correlation without this point is -0.78.

c) The regression line with the outlier is given by:

lm(formula = y ~ x)

Coefficients:
(Intercept) x
-494.46 -33.83

Thus, y=-494.46 -33.83x.

Now the regression line without the outlier is given by:

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept) x
-1368.63 -75.33

Thus, y=-1368.63 -75.33x.

So, YES the outlier is also strongly influential for the regression line.

Thus, done

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