Question

0.12 Roller coasters. The Roller Coaster Database (redb.com) contains lots of information about roller coasters all over the world. The following statistical model for predicting the top speed (in miles per hour) of a coaster was based on more than 100 roller coasters in the United States and data displayed on the database in November 2003: TopSpeed = 54 + 7.6TypeCode + E where TypeCode = 1 for steel roller coasters and TypeCode 0 for wooden roller coasters. a. What top speed does this model predict for a wooden rolle coaster? b. What top speed does this model predict for a steel roller coaster? c. Determine the difference in predicted speeds in miles per hour for the two types of coasters. Also identify where this number appears in the model equation, and explain why that makes sense

Answer 1

The predicted model equation for the top speed of roller coasters can be written as

$\widehat{\text{TopSpeed}}=54+7.6\text{TypeCode}$

where $\widehat{\text{TopSpeed}}$ is the predicted value for Top speed. (We drop the random error term $\epsilon$ from this equation)

a) We know that a wooded roller coaster is indicated by $\text{TypeCode}=0$. We substitute this value in the equation above.

$\widehat{\text{TopSpeed}}=54+7.6\times 0=54$

ans: The top speed predicted by this model for a wooden roller coaster is 54 miles per hour

b) a steel roller coaster is indicated by $\text{TypeCode}=1$. We substitute this value in the equation above.

$\widehat{\text{TopSpeed}}=54+7.6\times 1=61.6$

ans: The top speed predicted by this model for a steel roller coaster is 61.6 miles per hour

c) The difference in predicted speeds for the 2 types of roller coasters is

$\begin{align*} (\text{predicted speed for steel roller coaster})-(\text{predicted speed for wooden roller coaster})=61.6-54=7.6 \end{align*}$

The difference in predicted speeds for the 2 types of roller coasters is 7.6 miles per hour

This value is the same as the slope coefficient of TypeCode in the equation, which is 7.6.

Since type code is a binary variable, taking the values 0 or 1, the slope coefficient should indicate the difference in the top speed when the typecode changes from 0 (wooden coaster) to 1 (steel coaster). Hence it makes sense that the slope coefficient for typecode would indicate the difference in predicted speeds for 2 types of roller coasters.