Many factors will influence the distance a paper airplane will fly. Weight and s

Question

Many factors will influence the distance a paper airplane will fly. Weight and shape are two factors. Through this project we will attempt to determine the optimal shape and weight for distance. We will also compare regression and ANOVA to better understand key differences between the methods. In your data, you have four groups and the groups will probably have different means. We will use ANOVA and regression to test whether these differences are statistically significant. The ANOVA model for this project is: y,,-+4 +6,, for i= 1,2,3,4 and js 1,2,3,4,5 where E,,-N(0,2) i,j It's possible that the 4 means will lie on a line. If so, a test that the slope is zero (Ho : = 0 ) will also test for a difference in the four means. A linear regression model may also be appropriate: yi-A +Ax+E, for i = 1,2, ,n where , ~N(0, *) For the data you collected n = 20 because you did 5 trials with each of 4 different models.

Explanation / Answer

We can use R to analysis the data

Input are

flight=c("LP", "LP","HB","HB","LP","HB","HB","HP","LP","HB","LP","HB","LB","HP","HP","HP","HP","LB","LP","LB","LB","LB")
X=c(7.4,8.1,8.9,4.8,7.5,7.2,9.8,12.7,9.1,8.11,10,13.7,5.3,9.3,10.2,9.1,9.1,12.11,11.11,9.1,9.5,8)
data=data.frame(flight, X)
levels(data$flight)
data$flight=ordered(data$flight,levels = c("HB", "HP" ,"LB", "LP"))
library("ggpubr")
ggboxplot(data, x = "flight", y = "X",
          color = "flight", palette = c("#00AFBB", "#E7B800", "#FC4E07","#FC4E07"),
          order = c("HB", "HP" ,"LB", "LP"),
          ylab = "X", xlab = "Treatment")

# Compute the analysis of variance
res.aov <- aov(X~flight, data = data)
# Summary of the analysis
summary(res.aov)

Output

summary(res.aov)
            Df Sum Sq Mean Sq F value Pr(>F)
flight       3   6.30   2.099   0.425 0.737
Residuals   18 88.85   4.936             
>

L=lm(X~flight, data = data)
summary(L)

Call:
lm(formula = X ~ flight, data = data)

Residuals:
    Min      1Q Median      3Q     Max
-3.9517 -0.9800 -0.2608 0.9608 4.9483

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)   9.1255     0.4757 19.185 1.97e-13 ***
flight.L     -0.2075     0.9161 -0.227    0.823  
flight.Q     -0.6310     0.9513 -0.663    0.516  
flight.C      0.8834     0.9853   0.897    0.382  
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.222 on 18 degrees of freedom
Multiple R-squared: 0.06617,   Adjusted R-squared: -0.08947
F-statistic: 0.4251 on 3 and 18 DF, p-value: 0.7373

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