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,,-μ+4 +6,, for i= 1,2,3,4 and js 1,2,3,4,5 where E,,-N(0,σ2) 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 ε, ~N(0, σ*) For the data you collected n = 20 because you did 5 trials with each of 4 different models.

Answer 1

Solution:-) We will use least squares regression line using R where response variable is height and independent variable is x. We have our output as

Note:- Upper side is R-Code and lower side is R-Output

Ya ! O source on Save Run Source y<-c (1:22) 2 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, 4 length(x) 5 summary(1n(y-x))| 5:17 Top Level) R Sc Console Terminal > yく-c (1:22) x<-c(7.4,8.1, 8.9, 13.7, 5.3 4.8, 9.1 ,12.7, 9.1, ,9.5,8) 7.2 9.8 8.11,10, 10.2, 9.1 12.1,11.1 > length(x) [1] 22 > summary(In(y~x)) Call 1m(formula -y ~x) Residuals: -8. 8026 -4.7040 -0. 9591 5. 0985 11.5969 coefficients: (Intercept) 2. 3959 Min 1Q Median 3Q Max Estimate std. Error t value Pr(ltI) 0. 695 0. 6448 1.552 0.136 6.0159 0. 398 1.0009 Residual standard error: 6. 286 on 20 degrees of freedom Multiple R-squared: 0.1075, Adjusted R-squared: 0.06291 F-statistic: 2.41 on 1 and 20 DF, p-value: 0.1363

We have the hypotheses for coefficient of slope as

$H_0 : \beta_1 =0 \\ H_1: \beta_1 \neq 0$

We get our t-statistic as 0.395 and corresponding p-value as 0.695.

At 5% level of significance we get p-value greater than 0.05. Hence there is statistically significant.

4) p-value for the ANOVA is 0.71 and p-value for regression is 0.71 which is appropriately same. Hence they both calculate the same p-value for the test of statistically significance.