Forty-six mountains in the Adirondacks of upstate New York are known as the High Peaks, with elevations near or above 4000 feet. A goal for hikers in the region is to become a “46er” by scaling each of these peaks. The data below contains information on the elevation (in ft) of each peak along with data on typical hikes including the ascent (in ft), round-trip distance (in mi), difficulty ration (on a scale from 1-7 where 7 is the most difficult), and expected trip time (in hr).
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data(HighPeaks)
HighPeaks %>% head## Peak Elevation Difficulty Ascent Length Time
## 1 Mt. Marcy 5344 5 3166 14.8 10.0
## 2 Algonquin Peak 5114 5 2936 9.6 9.0
## 3 Mt. Haystack 4960 7 3570 17.8 12.0
## 4 Mt. Skylight 4926 7 4265 17.9 15.0
## 5 Whiteface Mtn. 4867 4 2535 10.4 8.5
## 6 Dix Mtn. 4857 5 2800 13.2 10.0Create and interpret a scatter plot of \(Y=Time\) versus \(X_1 = Elevation\).
#ggplot(HighPeaks, aes(x=?, y=?)) +
# geom_point() +
# labs(title="Title?", x="x label?", y = "y label?")Answer: [Write your answer here]
Consider a MLR using \(X_1 = Elevation\) and \(X_2 = Length\) to predict \(Y=Time\). Assess the importance of each predictor together and separately.
mod1 <- lm(Time ~ Elevation, HighPeaks)
mod2 <- lm(Time ~ Length, HighPeaks)
mod3 <- lm(Time ~ Elevation + Length, HighPeaks)
## use this additional space to keep codingAnswer: [Write your answer here]
Construct an added variable plot to see the effect of adding \(X_1=Elevation\) to a model that contains only \(X_2=Length\) to predict \(Y=Time\). Does this plot indicate there is information in \(X_1\) that is useful for predicting \(Y\), given that \(X_2\) is included in the model? Are there any unsual points indicated in this added variable plot?
#added_varb_plot_data <- tibble(res1 = ?$residuals, res2 = ?$residuals)
#ggplot(added_varb_plot_data, aes(x=res2, y=res1)) +
# geom_point() +
# geom_abline(intercept=0) +
# geom_text(label=rownames(added_varb_plot_data), nudge_y = -5) +
# labs(title="Added variable plot", x="Residuals for regressing ? on ?",y="Residuals for regressing ? on ?")Answer: [Write your answer here]
Write the estimated regression equation for a MLR model of your choice for this data and report the adjusted \(R^2\) value and interpret the residuals vs fitted values plot.
#mymod <- lm(? ~ ?, HighPeaks)
#summary(mymod)$adj.r.squared
#mymod_data <- HighPeaks %>% mutate(resids = mymod$residuals,
# fits = mymod$fitted.values)
#ggplot(?, aes(x=?, y=?)) +
# geom_point() +
# labs(title="Residuals vs fitted values", x= "Fitted values", y="Residuals")Answer: [Write your answer here]
Calculate the studentized residuals for each data point in your model from Q4. Are there any data points that stand out as unusual? If so, identify these observation(s).
# mymod_data2 <- mymod_data %>% mutate(student_resids = rstudent(?))
# mymod_data2$student_resids %>% sort Answer: [Write your answer here]
Calculate the leverage for each data point in your model from Q4. Are there any values that are unduly high? If so, identify these observation(s).
# mymod_data2 <- mymod_data %>% mutate(leverage = hatvalues(?))
# mymod_data2$leverage %>% sort Answer: [Write your answer here]