Emufinal

library(s20x)
emu.df = read.table('emu.txt', header = TRUE)
summary(emu.df)

     height          weight      
 Min.   :1.009   Min.   : 27.02  
 1st Qu.:1.255   1st Qu.: 37.34  
 Median :1.519   Median : 45.01  
 Mean   :1.451   Mean   : 47.40  
 3rd Qu.:1.594   3rd Qu.: 50.83  
 Max.   :1.831   Max.   :108.86  

plot(weight ~ height, data = emu.df)
abline(a=-15.33, b=43.23, col="red")

Exploratory Analysis:

The mean emu height in this sample is 1.45 meters, and the mean weight is approximately 47.4 kilograms. There is a moderately strong linear relationship between the height and the weight of an emu, and the scatter is relatively constant. There is one unusual value, with a weight of over 100 kilograms, and a height of about 1.6 meters.

emu.fit = lm(weight ~ height, data = emu.df)

plot(emu.fit, which=1)

modcheck(emu.fit)

summary(emu.fit)

Call:
lm(formula = weight ~ height, data = emu.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.022  -8.309  -3.633   3.284  53.970 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)   -15.33      23.05  -0.665   0.5155  
height         43.23      15.69   2.756   0.0141 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.49 on 16 degrees of freedom
Multiple R-squared:  0.3219,    Adjusted R-squared:  0.2796 
F-statistic: 7.597 on 1 and 16 DF,  p-value: 0.01405

confint(emu.fit)

                 2.5 %   97.5 %
(Intercept) -64.194057 33.53649
height        9.980616 76.48495

plot(weight ~ height, main="Emu height vs weight", data = emu.df)
abline(emu.fit)

# Check for effect of dropping observation 11
emu.fit2 <- lm(weight ~ height, data = emu.df[-11, ])
summary(emu.fit2)

Call:
lm(formula = weight ~ height, data = emu.df[-11, ])

Residuals:
    Min      1Q  Median      3Q     Max 
-10.769  -4.606  -1.195   4.491  13.390 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -3.386      9.907  -0.342 0.737215    
height        32.741      6.786   4.825 0.000223 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.591 on 15 degrees of freedom
Multiple R-squared:  0.6082,    Adjusted R-squared:  0.582 
F-statistic: 23.28 on 1 and 15 DF,  p-value: 0.0002227

predheight.df = data.frame(height = 1.5)
predict(emu.fit, predheight.df, interval = "prediction")

       fit      lwr      upr
1 49.52039 15.74109 83.29969

Methods and Assumptions Check:

Given the scatter is relatively consistent, a linear model was fitted. The data appears to be independently collected. The equality of variance and normality assumptions are met. While point 11 had a significance greater than 0.4 in the Cooks plot, when it was removed, the change was less than one standard error, so this point was in the final analysis as it did not have undue influence. Our final model is \(weight = \beta_0 + \beta_1 × height+ \epsilon_i\) where \(\overset{\mathrm{iid}}{\sim} N(0, \sigma^2)\).

Executive Summary:

The aim is to investigate the relationship between height and weight of emus, and estimate the weight of a specific emu. We found that there is a positive correlation between the height and weight of an emu. We carried out a predication for the weight of an emu with the height of 1.5 meters, and it gave us a range of 15.7 to 83.3 kilograms. Our model can explain 31% of the variation.