Question

temperatures at different times of day are shown. how can the relationship between temperature and time be described? would a linear model be a good fit for the data? explain
the temperature generally as the time after midnight increases. a linear model is best for data that have a rate of change, so a linear model a good fit for the data.

Explanation:

Looking at the temperatures: 12 AM ($67^{\circ}F$), 4 AM ($56^{\circ}F$), 8 AM ($60^{\circ}F$), 12 PM ($76^{\circ}F$), 4 PM ($79^{\circ}F$), 8 PM ($71^{\circ}F$), 12 AM ($67^{\circ}F$). As time after midnight (12 AM) increases (from 12 AM - 4 AM - 8 AM - 12 PM - 4 PM - 8 PM - 12 AM), the temperature first decreases (12 AM - 4 AM), then increases (4 AM - 4 PM), then decreases again (4 PM - 12 AM). A linear model requires a constant rate of change (either always increasing at a constant slope or always decreasing at a constant slope). Since the temperature does not have a constant rate of change (it changes direction - up and down), a linear model is not suitable.

Answer:

The temperature generally changes direction (first decreases, then increases, then decreases) as the time after midnight increases. A linear model is best for data that have a constant rate of change, so a linear model is not a good fit for the data.