Mark Breunig

WATR 784

Lab 2 – “Math Stats”

Sparrow Wing Length Linear Model

1)

2)

Numerical / Trial and Error
Slope / 0.27 / Slope / RSS
Intercept / 0.72 / 0 / 19.66
RSS / 0.52 / 0.1 / 8.12
0.2 / 1.82
0.3 / 0.76
0.4 / 4.94
0.5 / 20.23

2) Analytical

SUMMARY OUTPUT
Regression Statistics
Multiple R / 0.99
R Square / 0.97
Adjusted R Square / 0.97
Standard Error / 0.22
Observations / 13.00
ANOVA
df / SS / MS / F / Significance F
Regression / 1.00 / 19.13 / 19.13 / 401.09 / 0.00
Residual / 11.00 / 0.52 / 0.05
Total / 12.00 / 19.66
Coefficients / Standard Error / t Stat / P-value / Lower 95% / Upper 95% / Lower 95.0% / Upper 95.0%
Intercept / 0.71 / 0.15 / 4.82 / 0.00 / 0.39 / 1.04 / 0.39 / 1.04
X Variable 1 / 0.27 / 0.01 / 20.03 / 0.00 / 0.24 / 0.30 / 0.24 / 0.30
RESIDUAL OUTPUT
Observation / Predicted Y / Residuals / Standard Residuals
1 / 1.52 / -0.12 / -0.59
2 / 1.79 / -0.29 / -1.41
3 / 2.06 / 0.14 / 0.65
4 / 2.33 / 0.07 / 0.31
5 / 2.87 / 0.23 / 1.08
6 / 3.15 / 0.05 / 0.26
7 / 3.42 / -0.22 / -1.03
8 / 3.69 / 0.21 / 1.03
9 / 3.96 / 0.14 / 0.69
10 / 4.50 / 0.20 / 0.97
11 / 4.77 / -0.27 / -1.27
12 / 5.04 / 0.16 / 0.78
13 / 5.31 / -0.31 / -1.47

3)

4) Based on this linear regression model, sparrow wing length (of newly born sparrows) increases at a constant rate. As each additional day progresses, wing length increases by approximately 0.27 cm. If one assumes a similar constant growth rate from birth to age of 3 days, than a sparrow from this population is born with a wing length of 0.72 cm.

5) The linear regression used in this analysis fits the data quite well. In this particular application, a RSS of 0.52 is very acceptable. An R2 value of 0.97 also indicates a very close fit. The maximum residual error is -0.31, not considering scale this is very acceptable. However, if one does consider scale, the model fails to accurately describe one point, as there is a 20% error at age 4 days. Since all other % errors are below 10%, the model is a good fit overall.

6) As mentioned in question 5, the model does indeed fit the data. The particular application of the model determines the acceptable error. Since the residual error in this model is not considered significant, it would be appropriate for most applications.

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