Question

The Marseille Water Taxi ferries tourists from the harbor at Marseille, France, to the Frioul Islands in the Mediterranean Sea. The table below shows the number of passengers on the noontime ferry over seven randomly selected days along with the current ambient temperature in degrees Celsius.

Temperature Passengers

16 15

19 20

22 20

26 22

18 10

24 18

Run a regression where "Temperature" is the independent variable and "Passengers" is the dependent variable. Interpret the coefficient of determination in your own words.

Answer 1

Sol:

create a dataframe df2 with the the given data

use lm fucntion in R to fit a model

Plot function to get the scatterplot

abline to get the regression line on plot

summary function in R to get R sq

Rcode:

levels = c("1", "2", "3","4"))
df2 =read.table(header = TRUE, text ="
Temperature Passengers

16 15

19 20

22 20

26 22

18 10

24 18

"
)
df2

linreg <- lm(Passengers~Temperature,data=df2)
plot(x= df2$Temperature,y=df2$Passengers,
main="Scatterplot of Passengers vs Temperature",
xlab="Temperature",ylab="Passengers")
abline(coef(linreg)[1:2],col='red')

## rounded coefficients for better output
cf <- round(coef(linreg), 4)

check to avoid having plus followed by minus for negative coefficients
eq <- paste0("Passengers = ", cf[1],
ifelse(sign(cf[2])==1, " + ", " - "), abs(cf[2]), "Temperature "
)

## printing of the equation
mtext(eq, 3, line=0)

Call:
lm(formula = Passengers ~ Temperature, data = df2)

Residuals:
1 2 3 4 5 6
1.249 3.922 1.595 0.492 -5.302 -1.957

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.3387 8.8956 0.150 0.888
Temperature 0.7757 0.4211 1.842 0.139

Residual standard error: 3.594 on 4 degrees of freedom
Multiple R-squared: 0.4589,   Adjusted R-squared: 0.3237
F-statistic: 3.393 on 1 and 4 DF, p-value: 0.1393

Regression line to predict passengers is

passengers=1.3387+0.7757 *temperature

R sq=0.4589

45.89% variation in Passengers is explained by model

Explained variance=45.89%

Unexplained variance=100-45.89=54.11%