Normally, Calliope would measure time in years since 1995. But she recently learned about the point-slope equation of a line, she is wondering what it would look like if she just used \(Y\) for the actual year instead. That is, the variables would be
\begin{align*}
Y \amp= \text{ time (actual year) }\sim \text{ indep}\\
T \amp= \text{ average price of a movie ticket (\$) }\sim \text{ dep}
\end{align*}
Both the linear and exponential model did fairly well at predicting today’s prices. The linear model estimate of $11.29 was \(12.87-11.29 = \$1.58\) lower than the actual average price of a movie ticket but the exponential model estimate of $13.90 was about \(13.90-12.87 = \$1.03\) higher. The exponential is probably a slightly better estimate, but there’s some rounding error here so Calliope decides they are both quite close.
My parents bought the house I grew up in for $35,000 in 1964 and sold it 40 years later (in 2004) for $342,000. True story. (It was before the housing bubble burst.)
Write a linear and exponential model and compare the house’s value in 2026 according to each model. Don’t forget to name the variables, including units. Use the actual year.
The number of manufacturing jobs in the state has been declining for decades. In 1970, there were 1.2 million such jobs in the state but by 2010 there were only 0.6 million such jobs.
Sales of hybrid cars in the United States have continued to increase. In 1999, 17 (yes, seventeen) hybrid cars were sold. By 2002, that number was up to 34,521 hybrid cars sold. Compare what the linear and exponential models projected for hybrid car sales in the year 2025 and compare to the actual number which is reported to be 2.05 million in 2025, which doesn’t even include electric cars.
