- We identified the trend using linear regression on the provided data points.
- The equation of the line of best fit is calculated to be approximately \(y = 0.1868x - 367.952\).
- Using this model, we set \(y = 15\) to solve for the year \(x\).
- The average movie ticket price is projected to reach \(\$15\) in the year \(\boxed{2050}\).
Explanation
- Data Analysis and Method Selection
To determine the line of best fit, we start by organizing our data for the years and their corresponding ticket prices. Let \(x\) represent the year and \(y\) represent the ticket price. The given data points are: (1990, 4.22), (1995, 4.35), (2000, 5.39), (2005, 6.41), (2010, 7.89), and (2015, 8.43). We will use the method of least squares to find the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Calculating the Line of Best Fit
We compute the necessary sums to determine the slope \(m\) and intercept \(b\).
- Number of data points \(n = 6\)
- Sum of years \(\sum x = 11990\)
- Sum of prices \(\sum y = 36.69\)
- Sum of squares of years \(\sum x^2 = 23960550\)
- Sum of products of years and prices \(\sum xy = 73507.55\)
Using the formula for the slope \(m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}\):
\(m = \frac{6 \times 73507.55 - 11990 \times 36.69}{6 \times 23960550 - 11990^2} = \frac{441045.3 - 439913.1}{143763300 - 143760100} = \frac{1132.2}{3200} \approx 0.1868\)
Now, calculate the intercept \(b = \bar{y} - m\bar{x}\):
First, the mean \(\bar{x} = \frac{11990}{6} \approx 1998.3333\)
Then, the mean \(\bar{y} = \frac{36.69}{6} = 6.115\)
\(b = 6.115 - (0.1868 \times 1998.3333) \approx -367.952\)
The equation for the line of best fit is approximately \(y = 0.1868x - 367.952\).
- Estimating the Future Year
Now, we estimate the year when the average ticket price reaches \(\$15\). We set \(y = 15\) in our linear equation and solve for \(x\):
\(15 = 0.1868x - 367.952\)
\(15 + 367.952 = 0.1868x\)
\(382.952 = 0.1868x\)
\(x = \frac{382.952}{0.1868} \approx 2050.06\)
Rounding to the nearest whole year, the price will reach \(\$15\) in the year \(2050\).
Examples
Linear regression is a powerful tool for predicting future outcomes based on historical data. Beyond predicting movie ticket prices, it is used by businesses to forecast demand for a product, by scientists to model the relationship between variables in experiments, and by urban planners to estimate future population growth based on historical trends. Mastering these calculations allows you to create data-driven projections in almost any field involving trends over time.