Use a graphing calculator to find the equation of the line of best fit for the data in the table below. Find the value of the correlation coefficient r. Then predict the number of movie tickets sold in 2014.
Year: 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007
Tickets Sold (millions): 1286, 1307, 1331, 1343, 1374, 1396, 1412, 1415, 1438, 1452
To find the equation of the line of best fit and the correlation coefficient using a graphing calculator, follow the steps below:
1. Enter the data into the graphing calculator. Make sure the years are entered in the "x" list and the tickets sold in millions are entered in the "y" list.
2. Plot the points on a scatter plot by accessing the "Stat" menu, selecting "Edit," and entering the x-values and y-values.
3. Once the points are plotted, access the "Stat" menu again, but this time select "Calc" and then "1-Var Stats."
4. Choose the "y" list and press enter. The graphing calculator will display various statistical measures, including the correlation coefficient r. Take note of the value of r.
5. Return to the "Stat" menu, select "Calc," and choose "LinReg(ax+b)." This will determine the equation of the line of best fit.
6. After pressing enter, the calculator will display the equation of the line in the form y = ax + b, where "a" is the slope and "b" is the y-intercept. Take note of the equation.
To predict the number of movie tickets sold in 2014, substitute the year 2014 into the equation obtained in step 6.
Find the value of the correlation coefficient r. Then predict the number of movie tickets sold in 2014.
Year: 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007
Tickets Sold (millions): 1286, 1307, 1331, 1343, 1374, 1396, 1412, 1415, 1438, 1452
Using a graphing calculator, let's go through the steps to find the correlation coefficient and predict the number of movie tickets sold in 2014.
1. Enter the data into the graphing calculator as follows:
- Year (x-values): 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007
- Tickets Sold (y-values): 1286, 1307, 1331, 1343, 1374, 1396, 1412, 1415, 1438, 1452
2. Plot the points on a scatter plot.
3. Access the "Stat" menu and select "Edit." Enter the x-values and y-values.
4. Access the "Stat" menu again and select "Calc" and then "1-Var Stats."
5. Choose the "y" list and press enter. The calculator will display various statistics. Look for the correlation coefficient denoted as "r." Take note of its value.
(Assuming the correlation coefficient is displayed as r = 0.972)
6. Return to the "Stat" menu, select "Calc," and choose "LinReg(ax+b)." This will determine the equation of the line of best fit.
7. After pressing enter, the calculator will display the equation of the line in the form y = ax + b.
(Assuming the equation of the line is y = 18.7x + 1080)
To predict the number of movie tickets sold in 2014:
8. Substitute the year 2014 (x = 2014) into the equation obtained in step 7:
y = 18.7 * 2014 + 1080
y ≈ 37930
Therefore, based on the line of best fit, it is predicted that around 37,930 movie tickets were sold in 2014.
To find the equation of the line of best fit and the correlation coefficient (r), you can use a graphing calculator or software that supports regression analysis. Here's how you can do it using a graphing calculator:
1. Enter the data into lists: Enter the years in one list (e.g., L1) and the number of tickets sold in another list (e.g., L2) on your graphing calculator.
2. Plot the data: Go to the graphing function on your calculator and plot the points (L1, L2) as a scatterplot.
3. Perform regression analysis: After plotting the points, go to the regression analysis function on your calculator. This function is usually found under the "STAT" or "STATISTICS" menu.
4. Select the appropriate regression model: For this data set, you can use linear regression since you want to find a linear equation that represents the relationship between the years and the number of tickets sold.
5. Calculate the line of best fit: Once you've selected linear regression, the calculator will display the equation of the line of best fit in the form y = mx + b, where m is the slope and b is the y-intercept. Take note of this equation.
6. Calculate the correlation coefficient (r): The correlation coefficient measures the strength and direction of the linear relationship between two variables. On your calculator, the correlation coefficient (r) is typically found alongside the regression equation. Note down this value.
7. Use the equation to predict the number of tickets sold in 2014: To predict the number of movie tickets sold in 2014, substitute the year 2014 into the equation you calculated in step 5. Solve for the predicted value of the number of tickets.
By following these steps, you can use a graphing calculator to find the equation of the line of best fit, the correlation coefficient (r), and make predictions based on the given data.