Question

An archaeologist measures the concentration of nitrogen (g/100 g of bone) in 41 human skeletons that have been interred for varying numbers of years. He is interested in how nitrogen concentration varies depending on length of interment. The data are available in the file nitro.csv.

If you have access to Jupyter R Notebooks please show work there!

a. Give the equation of the fitted regression line.

b. Create a plot of nitrogen concentration vs. length of interment, including the fitted regression line.

c. Interpret the slope of the fitted regression line in the language of the problem.

d. State the R2 value and its interpretation (in the language of the problem).

e. The archaeologist finds an additional skeleton that was interred for 80 years. What is the best prediction of its nitrogen concentration?

file:

"Intern.Time","Nitrogen"

88.5,3.6
85.2,3.53
71.8,3.7
70.6,3.82
68,3.58
71.6,3.61
70.2,4.01
55.5,3.74
36.5,3.98
36.3,3.86
46.5,3.88
35.9,3.95
45.5,4.11
43,3.77
44.9,3.82
59.5,3.79
58.3,3.74
56.5,3.93
56.3,3.78
43,4.01
42.5,4.17
29,3.82
35.3,4.11
93.6,3.52
90,3.63
88,3.84
84.4,3.69
84,3.55
79.7,3.53
67.4,3.67
64.7,3.74
64.7,3.62
38.3,3.79
59.6,3.82
32,4.02
32.2,4.05
26.5,4.01
34.7,3.76
27.6,3.87
35.7,3.82
49.6,3.65

Answer 1

a) The researcher wants to predict the concentration of nitrogen using the year of Intern Time.

Dependent variable =  Nitrogen concentration

Independent variable =Intern.Time

Fitted Regression Model in R:

> Model = lm( Nitrogen concentration~Intern.Time,data = data)
> Model

lm(formula = Nitrogen ~ Intern.Time, data = data)

Coefficients:
(Intercept) Intern.Time  
4.139659 -0.006008  

so fitted model from the above data is,

Nitrogen = 4.139659 - 0.006008 * Intern.Time

from the above model, we can say that if the Intern. Time increase by one unit then the concentration of nitrogen decreases by 0.006008  

b) The regression line plot is,

From the above graph, the trend is downward i.e. if the volume of Nitrogen increases then Intern. Time decreases.

c) From the above model, from the above model, we can say that if the Intern. Time increase by one unit then the concentration of nitrogen decreases by 0.006008  

d) The summary of the model:

Coefficients Estimate Std. Error t value Pr(>|t|)   

(Intercept) 4.1396588 0.0591916 69.937 < 0.0000000000000002 ***

Intern.Time -0.0060076 0.0009933 -6.048 0.000000444 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1265 on 39 degrees of freedom

Multiple R-squared: 0.484, Adjusted R-squared: 0.4708

F-statistic: 36.58 on 1 and 39 DF, p-value: 0.0000004438

From the above summary, the value of Adjusted R^2 = 0.4708

i.e.Model is not effective

Because the value of R^2 is higher then the accuracy of the model is also increases.

if R^2 > 0.80 then model very good.

e) The skeleton that was interred for 80 years

So from the model,

Nitrogen concentration = 4.139659 - 0.006008 * 80

= 3.659019

So the Nitrogen concentration at 80 years age 3.66 (g/100 g of bone)

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