The Siege Setup

We performed a designed experiment on the effect of Release Angle on the distance in which a ball is thrown (measured at its furthest distance). Specifically, we studied whether the settings of 175, 180, and 185 degrees significantly differ in their mean distance thrown. Since pulling the lever back takes additional work, we investigated whether this made a significant difference on the mean distance thrown. The other factors were set to constant values of the following:

  Fire Angle = 90°
  Bungee Position = 200mm
  Pin Elevation = 200mm
  Cup Elevation = 300mm

To test this hypothesis, we used a completely randomized design with an α = 0.05.

Sample Size

To save resources during the preparation for the siege, we determined how many samples should be collected to detect a mean difference with a medium effect (i.e. 50% of the standard deviation) with a probability of 75%.

power1 = 0.75
std1   = sd(effectiveReleaseAngle)^2
sampleNumber <- power.anova.test(n = NULL,
                 groups = length(effectiveReleaseAngle),
                 between.var = var(effectiveReleaseAngle),
                 within.var = std1,
                 sig.level = alpha,
                 power = power1)

The results of the sample size calcuation showed the need to run 6 launches per Release Angle.

Trial Run Order

We knew that in order to reduce the margin of error in our sampling process, we would randomize our design for collecting catapult data. In order to achieve this we leveraged the built-in capabilities of r through the design.crd function.

design <- design.crd(trt = effectiveReleaseAngle,
                     r = ceiling(sampleNumber$n),
                     seed = 1234)

The final randomized run order is as follows:
  185, 180, 185, 180, 180, 185, 175, 175, 175, 180, 180, 175, 175, 175, 185, 185, 180, 185

Trial Runs

We then ran the trials on the Statapult and collected the following distance measurements:

Release Angle
175° 180° 185°
1 258 (7) 276 (2) 281 (1)
2 255 (8) 266 (4) 278 (3)
3 271 (9) 286 (5) 289 (6)
4 270 (12) 270 (10) 265 (15)
5 270 (13) 276 (11) 276 (16)
6 264 (14) 285 (17) 274 (18)
All values are distances in mm & (run order)

Hypothesis Testing

We needed to then understand if Release Angle plays a significant role in the distance the ball travels. To complete this task, we ran a One-way ANOVA test.

Hypotheses

The hypotheses for the ANOVA were as follows:
 Null:     H0: μ 175 = μ 180 = μ 185
 Alternate: H1: μi ≠ μj for at least one pair (i,j)

ANOVA

The ANOVA was then run in R.

dat <- data.frame(angles   = design$book$effectiveReleaseAngle,
                  distance = recordedDistance)
dataov <- aov(distance~angles,data = dat)
Angle_P_Value <- round(summary(dataov)[[1]][1, 5], digits= 2)

Which showed that at least one of the means was significantly different than the others.

##             Df Sum Sq Mean Sq F value Pr(>F)  
## angles       2  593.4  296.72   5.142 0.0199 *
## Residuals   15  865.7   57.71                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We can see that the p-value < 0.02 , and with an α = 0.05, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between at least one of the three sample mean Release Angle launch distances.

Residuals

The residuals were then assessed for normality and constant variance. From the charts below the residuals are indeed homoscedastic and normal.

autoplot(dataov) + geom_point(shape = 19, size = 2, alpha = 0.6, color = "#fbad00")

Pairwise Comparisons

Pairwise comparisons were conducted to understand which pairs were statistically significantly different. The following hypotheses were created for a Tukey’s Test.
 Null:     H0: μi = μj
 Alternate: H1: μi ≠ μj for all i ≠ j.

The Tukey’s Test was then run in R.

datHSD <- TukeyHSD(dataov)

With the following results:

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = distance ~ angles, data = dat)
## 
## $angles
##               diff         lwr      upr     p adj
## 180-175 11.8333333   0.4408168 23.22585 0.0413032
## 185-175 12.5000000   1.1074834 23.89252 0.0308154
## 185-180  0.6666667 -10.7258499 12.05918 0.9873545

We can see that the pairwise comparisons resulted in two of the three comparisons being significant to an α = 0.05.

Release Angle Comparison P-value
180 - 175 0.041 *
185 - 175 0.031 *
185 - 180 0.988
" * " denotes a significant p-value

For the ‘180-175’ and ‘185-175’ comparisons, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between the sample mean distances.

For the ‘185-180’ comparison, we fail to reject the Null hypothesis. There is not enough evidence to support the claim of a significant difference between the sample mean distances.

Conclusions & Recommendations

Release Angle was shown to be a significant factor for ball travel distances for both 175° and 180°. It was not shown to be significant when pulled beyond 180° to the 185° position.

The recommendation is to not pull back beyond 180° because the extra effort isn’t resulting in greater distance.

It is also recommended to study other Release Angles to understand the impact they play on ball distance traveled throughout the range of arm motion.