case1001: Galileo's Data on the Motion of Falling Bodies

In Sleuth3: Data Sets from Ramsey and Schafer's "Statistical Sleuth (3rd Ed)"

Description

In 1609 Galileo proved mathematically that the trajectory of a body falling with a horizontal velocity component is a parabola. His search for an experimental setting in which horizontal motion was not affected appreciably (to study inertia) let him to construct a certain apparatus. The data comes from one of his experiments.

Usage

case1001

Format

A data frame with 7 observations on the following 2 variables.

Distance

horizontal distances (in punti)

Height

initial height (in punti)

Source

Ramsey, F.L. and Schafer, D.W. (2013). The Statistical Sleuth: A Course in Methods of Data Analysis (3rd ed), Cengage Learning.

Examples

str(case1001)
attach(case1001)
      
## EXPLORATION
plot(Distance ~ Height)
myLm <- lm(Distance ~ Height)
plot(myLm, which=1)         
height2 <- Height^2
myLm2 <- lm(Distance ~ Height + height2)
plot(myLm2, which=1)        
summary(myLm2) # Note p-value for quadratic term (it's small)
height3 <- Height^3
myLm3 <- update(myLm2, ~ . + height3)  
plot(myLm3,which=1)         
summary(myLm3) # Note p-value for cubic term (it's small)              
height4 <- Height^4
myLm4 <- update(myLm3, ~ . + height4)
summary(myLm4) # Note p-value for quartic term (it's not small)             

## DISPLAY FOR PRESENTATION 
plot(Distance ~ Height, xlab="Initial Height (Punti)",
  ylab="Horizontal Distance Traveled (Punti)",
  main="Galileo's Falling Body Experiment",
  pch=21, bg="green", lwd=2, cex=2)
dummyHeight     <- seq(min(Height),max(Height),length=100)         
betaQ           <- myLm2$coef  
quadraticCurve  <- betaQ[1] + betaQ[2]*dummyHeight + betaQ[3]*dummyHeight^2  
lines(quadraticCurve ~ dummyHeight,col="blue",lwd=3)  
betaC           <- myLm3$coef # coefficients of cubic model  
cubicCurve      <- betaC[1] + betaC[2]*dummyHeight + betaC[3]*dummyHeight^2 +
  betaC[4]*dummyHeight^3  
lines(cubicCurve ~ dummyHeight,lty=3,col="red",lwd=3) 
legend(590,290,legend=c(expression("Quadratic Fit  "*R^2*" = 99.0%"),
  expression("Cubic Fit        "*R^2*" = 99.9%")),  
  lty=c(1,3),col=c("blue","red"), lwd=c(3,3))
  
detach(case1001)

Example output

'data.frame':	7 obs. of  2 variables:
 $ Distance: int  253 337 395 451 495 534 573
 $ Height  : int  100 200 300 450 600 800 1000

Call:
lm(formula = Distance ~ Height + height2)

Residuals:
      1       2       3       4       5       6       7 
-14.308   9.170  13.523   1.940  -6.177 -12.607   8.458 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.999e+02  1.676e+01  11.928 0.000283 ***
Height       7.083e-01  7.482e-02   9.467 0.000695 ***
height2     -3.437e-04  6.678e-05  -5.147 0.006760 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 13.64 on 4 degrees of freedom
Multiple R-squared:  0.9903,	Adjusted R-squared:  0.9855 
F-statistic:   205 on 2 and 4 DF,  p-value: 9.333e-05


Call:
lm(formula = Distance ~ Height + height2 + height3)

Residuals:
       1        2        3        4        5        6        7 
-2.40359  3.58091  1.89175 -4.46885 -0.08044  2.32159 -0.84138 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.558e+02  8.326e+00  18.710 0.000333 ***
Height       1.115e+00  6.567e-02  16.983 0.000445 ***
height2     -1.245e-03  1.384e-04  -8.994 0.002902 ** 
height3      5.477e-07  8.327e-08   6.577 0.007150 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.011 on 3 degrees of freedom
Multiple R-squared:  0.9994,	Adjusted R-squared:  0.9987 
F-statistic:  1595 on 3 and 3 DF,  p-value: 2.662e-05


Call:
lm(formula = Distance ~ Height + height2 + height3 + height4)

Residuals:
      1       2       3       4       5       6       7 
-0.4433  0.9338  0.2576 -2.3092  2.3183 -0.9279  0.1708 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  1.383e+02  9.066e+00  15.254  0.00427 **
Height       1.346e+00  1.061e-01  12.690  0.00615 **
height2     -2.117e-03  3.793e-04  -5.582  0.03063 * 
height3      1.766e-06  5.186e-07   3.406  0.07644 . 
height4     -5.610e-10  2.375e-10  -2.362  0.14201   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.523 on 2 degrees of freedom
Multiple R-squared:  0.9998,	Adjusted R-squared:  0.9995 
F-statistic:  3024 on 4 and 2 DF,  p-value: 0.0003306

Sleuth3 documentation built on May 2, 2019, 6:41 a.m.