Données

vitesse <- c(4, 7, 8, 9, 10, 11, 11, 12, 12, 13,
             14, 15, 15, 16, 17, 18, 19, 20, 24, 25)

distance <- c(2, 4, 16, 10, 26, 17, 28, 20, 28, 26,
              36, 26, 54, 40, 50, 76, 46, 48, 92, 85)

1-1) Moyennes

mean_vitesse <- mean(vitesse)
mean_distance <- mean(distance)

mean_vitesse
## [1] 14
mean_distance
## [1] 36.5

1-2) Variances

var_vitesse <- var(vitesse)
var_distance <- var(distance)

var_vitesse
## [1] 29.78947
var_distance
## [1] 642.7895

1-3) Régression distance ~ vitesse

reg_d_sur_v <- lm(distance ~ vitesse)
summary(reg_d_sur_v)
## 
## Call:
## lm(formula = distance ~ vitesse)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.7968  -6.3048  -0.2032   5.6325  22.3127 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -23.655      6.230  -3.797  0.00132 ** 
## vitesse        4.297      0.416  10.329 5.42e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.897 on 18 degrees of freedom
## Multiple R-squared:  0.8556, Adjusted R-squared:  0.8476 
## F-statistic: 106.7 on 1 and 18 DF,  p-value: 5.416e-09

1-4) Régression vitesse ~ distance

reg_v_sur_d <- lm(vitesse ~ distance)
summary(reg_v_sur_d)
## 
## Call:
## lm(formula = vitesse ~ distance)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8657 -1.4578  0.1883  1.2900  3.7100 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.73168    0.84980   7.921 2.82e-07 ***
## distance     0.19913    0.01928  10.329 5.42e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.131 on 18 degrees of freedom
## Multiple R-squared:  0.8556, Adjusted R-squared:  0.8476 
## F-statistic: 106.7 on 1 and 18 DF,  p-value: 5.416e-09

1-5) Graphique + angle

plot(
  vitesse, distance, pch = 19, col = "blue",
  xlab = "Vitesse (mph)", ylab = "Distance (ft)",
  main = "Régressions croisées"
)

abline(reg_d_sur_v, col = "red", lwd = 2)

a <- coef(reg_v_sur_d)[1]
b <- coef(reg_v_sur_d)[2]
abline(a = -a / b, b = 1 / b, col = "green", lwd = 2)

legend("topleft",
       legend = c("D ~ V", "V ~ D inversée"),
       col = c("red", "green"), lty = 1, lwd = 2)

m1 <- coef(reg_d_sur_v)[2]
m2 <- 1 / coef(reg_v_sur_d)[2]

theta_deg <- atan(abs((m2 - m1) / (1 + m1 * m2))) * 180 / pi
theta_deg
## distance 
##  1.83914

1-6) Coefficient de corrélation linéaire

r <- cor(vitesse, distance)
r
## [1] 0.9250052

Conclusion Le coefficient de corrélation est élevé, indiquant une forte dépendance linéaire positive entre la vitesse et la distance de freinage.