M425P1 <- M425P %>%
filter(AttendedAlmostAlways == "Y", ClassActivitiesCompletedPerfectly == "Y", SkillsQuizzesCompletedPerfectly == "Y", MagicTwoGroups == 2,
)
M425P2 <- M425P %>%
filter(AttendedAlmostAlways == "Y", ClassActivitiesCompletedPerfectly == "Y", SkillsQuizzesCompletedPerfectly == "Y", MagicTwoGroups == 2, Midterm != 52
)library(ggplot2)
library(plotly)
midpred.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3), data = M425P2)
b <- coef(midpred.lm)
predintv <- predict(midpred.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x + b[3]*x^2 + b[4]*x^3, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_x = -11.25, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.009, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))midpred.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3), data = M425P2)
pander(summary(midpred.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 165.3 | 78.66 | 2.101 | 0.04382 |
| Midterm | -6.294 | 3.941 | -1.597 | 0.1204 |
| I(Midterm^2) | 0.1066 | 0.06157 | 1.732 | 0.0933 |
| I(Midterm^3) | -0.0005145 | 0.0003048 | -1.688 | 0.1015 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 9.366 | 0.5303 | 0.4849 |
I have selected the cubic model because although the R^2 and RSE is better in the “quinto” and “sexto” models in the appendix, they are also unreasonable. the curves are disproportionate to reality.
\underbrace{53.62}_\text{Final Exam Score} = {165.3} + \underbrace{X_{i}}_\text{Midterm Score} + {-6.294}\underbrace{X_{i}^2}_\text{I(Midterm Score^2)} + {0.1066} \underbrace{X_{i}^3}_\text{I(Midterm Score^3)} + {-0.0005145} + \epsilon_i
par(mfrow=c(1,3))
plot(midpred.lm, which=1)
qqPlot(midpred.lm$residuals, main="Q-Q Plot", col="cadetblue4", col.lines="seagreen", pch= 19, id=FALSE)
plot(midpred.lm$residuals, main = "Residuals vs. Order")
Everything looks fairly good.
There are 5 associated assumptions to the linear regression. All but #4 can be checked with these above plots.
Assumption 1: Linear Relationship Between X and Y
The linearity as shown in the residuals/fitted plot is a little wonky but it evens out. This is due to the dots being on the right side with more linearity than the left side.
Assumption 2: Normal Distribution of Error Terms
Most of the dots as shown in the Q-Q plot fall on or close to the line, there are a few at the top that vary from normality but this isn’t too bad.
Assumption 3: Constant Variance (X values)
We do see a push of the data to higher values. The right side is more occupied that the left. Constant variance is put into question here.
Assumption 4: Fixed X
The X_i values can be considered fixed and measured without error.
Assumption 5: Independent Error Terms
The Residuals vs Order plot shows the chaos in the data. Error terms do not appear to be correlated. There is not a clear pattern in the plot.
Assumption Conclusion
There is not sufficient evidence shown in the above plots to discredit the whole linear regression analysis.
library(pander)
# Prediction Interval
predintv <- predict(midpred.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 68.49 | 48.43 | 88.55 |
# Confidence Interval
confintv <- predict(midpred.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 68.49 | 62.37 | 74.6 |
The big question is then what if I get exactly my prediction? What will my final score be?
pander(paste(0.7*68.49 + 0.3*68))68.343
The prediction intervals:
High (0.3 x 68 + 0.7 x 88.55)
pander(paste(+ 0.3*68 + 0.7*88.55))82.385
Low (0.3 x 68 + 0.7 x 48.43)
pander(paste(0.3*68 + 0.7*48.43))54.301
If I go based off my final exam prediction I am likely to get a 68 but accounting for prediction intervals, I have a range of plus or minus 20 points. I believe I will be more confident in my final exam score with the addition of my midterm. 68 wasn’t too bad to start out as the midterm. I have the chance to get an 88 on the final, bringing my overall exam score to an 82 but there’s also a chance to get a 48 on the final and that will bring my overall exam score to a 54.
I am deciding to keep my midterm and shoot for a higher score than 68 on my final exam. I feel confident that I can aim for a 76. If I got a 76 on the final my overall exam grade would be a 74.
First of all: When examining the original residulas plots as well as the graph, there was an extreme outlier. Someone who got a 52 on the midterm and a 92 on the final exam. It was pulling the data away like crazy. It is number 21 as shown below. I removed that dot from the analysis. It made the R^2 for all models significantly better.
midpredd.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4), data = M425P1)
par(mfrow=c(1,3))
plot(midpredd.lm, which=1)
qqPlot(midpredd.lm$residuals, main="Q-Q Plot", col="cadetblue4", col.lines="seagreen", pch= 19, id=FALSE)
plot(midpredd.lm$residuals, main = "Residuals vs. Order")
How about if I get the high end of my confidence interval?
pander(paste(0.7*74.6 + 0.3*68))72.62
The low end?
pander(paste(0.7*62.37 + 0.3*68))64.059
The R^2 was low for this one so I moved to a quadratic:
midpred1.lm <- lm(FinalExam ~ Midterm, data = M425P2)
pander(summary(midpred1.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 26.14 | 9.921 | 2.635 | 0.01273 |
| Midterm | 0.6671 | 0.1203 | 5.543 | 3.712e-06 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 9.532 | 0.4822 | 0.4665 |
library(ggplot2)
library(plotly)
midpred1.lm <- lm(FinalExam ~ Midterm, data = M425P2)
b <- coef(midpred1.lm)
predintv <- predict(midpred1.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred1.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_x = -11.25, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.009, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))predintv1 <- predict(midpred1.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv1))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 71.5 | 51.56 | 91.44 |
# Confidence Interval
confintv1 <- predict(midpred1.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv1))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 71.5 | 66.87 | 76.13 |
The R^2 was almost worse for this one so I moved to a cubic:
midpred2.lm <- lm(FinalExam ~ Midterm + I(Midterm^2), data = M425P2)
pander(summary(midpred2.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 40.68 | 27.87 | 1.46 | 0.1541 |
| Midterm | 0.2291 | 0.7926 | 0.2891 | 0.7744 |
| I(Midterm^2) | 0.003103 | 0.005549 | 0.5592 | 0.5799 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 9.633 | 0.4872 | 0.4551 |
library(ggplot2)
library(plotly)
midpred2.lm <- lm(FinalExam ~ Midterm + I(Midterm^2), data = M425P2)
b <- coef(midpred2.lm)
predintv <- predict(midpred2.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred2.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x + b[3]*x^2, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_x = -11.25, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.009, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))predintv2 <- predict(midpred2.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv2))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 70.6 | 50.17 | 91.04 |
# Confidence Interval
confintv2 <- predict(midpred2.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv2))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 70.6 | 64.9 | 76.31 |
The adjusted R^2 was getting worse and the R^2 was still low for this one so as I was deciding on the 4 beta model in my analysis I also tested what it would be with 5 and 6 betas:
This is when it gets crazy:
midpred4.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4), data = M425P2)
pander(summary(midpred4.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 243.1 | 414.8 | 0.5859 | 0.5623 |
| Midterm | -11.66 | 28.36 | -0.411 | 0.684 |
| I(Midterm^2) | 0.2351 | 0.6756 | 0.348 | 0.7303 |
| I(Midterm^3) | -0.001811 | 0.006793 | -0.2666 | 0.7916 |
| I(Midterm^4) | 4.706e-06 | 2.463e-05 | 0.191 | 0.8498 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 9.515 | 0.5309 | 0.4683 |
library(ggplot2)
library(plotly)
midpred4.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4), data = M425P2)
b <- coef(midpred4.lm)
predintv <- predict(midpred4.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred4.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x + b[3]*x^2 + b[4]*x^3 + b[5]*x^4, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_y = -10, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.01, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))predintv4 <- predict(midpred4.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv4))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 68.79 | 48.13 | 89.45 |
# Confidence Interval
confintv4 <- predict(midpred4.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv4))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 68.79 | 61.78 | 75.81 |
midpred5.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4) + I(Midterm^5), data = M425P2)
pander(summary(midpred5.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | -1887 | 2539 | -0.7434 | 0.4632 |
| Midterm | 165.5 | 210.2 | 0.7873 | 0.4375 |
| I(Midterm^2) | -5.346 | 6.596 | -0.8105 | 0.4243 |
| I(Midterm^3) | 0.08258 | 0.09944 | 0.8305 | 0.4131 |
| I(Midterm^4) | -0.0006128 | 0.0007263 | -0.8437 | 0.4057 |
| I(Midterm^5) | 1.759e-06 | 2.067e-06 | 0.8507 | 0.4019 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 9.56 | 0.5423 | 0.4634 |
The R^2 is so much better! But the graph…
library(ggplot2)
library(plotly)
midpred5.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4) + I(Midterm^5), data = M425P2)
b <- coef(midpred5.lm)
predintv <- predict(midpred5.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred5.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x + b[3]*x^2 + b[4]*x^3 + b[5]*x^4 + b[6]*x^5, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_x = -11.25, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.009, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))The graph is basically saying that if you got a 40 on your midterm you are likely to get a 75 on the final. There aren’t any data points there so it is assuming that.
How about 6 betas?
predintv5 <- predict(midpred5.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv5))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 66.78 | 45.44 | 88.13 |
# Confidence Interval
confintv5 <- predict(midpred5.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv5))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 66.78 | 58.23 | 75.34 |
midpred6.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4) + I(Midterm^5) + I(Midterm^6), data = M425P2)
pander(summary(midpred6.lm))| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | -42644 | 16220 | -2.629 | 0.01375 |
| Midterm | 4090 | 1558 | 2.626 | 0.01386 |
| I(Midterm^2) | -156.4 | 59.81 | -2.615 | 0.0142 |
| I(Midterm^3) | 3.085 | 1.186 | 2.601 | 0.01468 |
| I(Midterm^4) | -0.0333 | 0.01289 | -2.583 | 0.01531 |
| I(Midterm^5) | 0.0001873 | 7.309e-05 | 2.562 | 0.01607 |
| I(Midterm^6) | -4.301e-07 | 1.694e-07 | -2.539 | 0.01696 |
| Observations | Residual Std. Error | R^2 | Adjusted R^2 |
|---|---|---|---|
| 35 | 8.771 | 0.628 | 0.5482 |
Again, the R^2 is so much better! But the graph is horrible.
I don’t think 120+ is possible on the final… this model is beyond reasonable probability.
library(ggplot2)
library(plotly)
midpred6.lm <- lm(FinalExam ~ Midterm + I(Midterm^2) + I(Midterm^3) + I(Midterm^4) + I(Midterm^5) + I(Midterm^6), data = M425P2)
b <- coef(midpred6.lm)
predintv <- predict(midpred6.lm, data.frame(Midterm = 68), interval = "prediction")
confintv <- predict(midpred6.lm, data.frame(Midterm = 68), interval = "confidence")
midterm.ggplot <- ggplot(M425P2, aes(y = FinalExam, x = Midterm, color = FinalExam)) +
geom_point(pch = 16, bg = "white", size = 3) +
stat_function(fun=function(x) b[1] + b[2]*x + b[3]*x^2 + b[4]*x^3 + b[5]*x^4 + b[6]*x^5 + b[7]*x^6, color = "seagreen3", size = 1.5) +
scale_color_gradient(low = "cadetblue1", high = "cadetblue4") +
labs(
title = "How Well Does My Midterm Predict My Final Exam?<br><sup>Let's Goooo<sup>",
x = "Past Midterm Scores (2019-2024)",
y = "Past Final Exam Scores (2019-2024)"
) +
ylim(min(M425P2$FinalExam) - 5, max(M425P2$FinalExam) + 5) +
theme_minimal() +
geom_point(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68))),
color = "firebrick", size = 4
) +
geom_text(
aes(x = 68, y = predict(midpred.lm, data.frame(Midterm = 68)), label = "Predicted Final Exam Score"),
color = "firebrick", nudge_x = -11.25, size = 3
) +
geom_segment(aes(x=68, xend=68, y=confintv[2], yend=confintv[3]),
alpha=0.04, color="goldenrod2", lwd=2) +
geom_segment(aes(x=68, xend=68, y=predintv[2], yend=predintv[3]),
alpha=0.009, color="firebrick", lwd=1.5)
ggplotly(midterm.ggplot, tooltip = c("x", "y"))predintv6 <- predict(midpred6.lm, data.frame(Midterm = 68), interval = "prediction")
pander(data.frame(`Prediction Interval` = predintv6))| Prediction.Interval.fit | Prediction.Interval.lwr | Prediction.Interval.upr |
|---|---|---|
| 64.78 | 45.11 | 84.46 |
# Confidence Interval
confintv6 <- predict(midpred6.lm, data.frame(Midterm = 68), interval = "confidence")
pander(data.frame(`Confidence Interval` = confintv6))| Confidence.Interval.fit | Confidence.Interval.lwr | Confidence.Interval.upr |
|---|---|---|
| 64.78 | 56.76 | 72.81 |
I have selected the cubic model because although the R^2 and RSE is better in the “quinto” and “sexto” models in the appendix, they are also unreasonable. the curves are disproportionate to reality.
: )