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
Find the equation of the line of best fit. Round to two decimal places as needed. Choose the correct answer below.
To find the correlation coefficient r and the equation of the line of best fit, we need to use regression analysis. Here is the solution:
Step 1: Calculate the mean of the x-values (years) and the mean of the y-values (tickets sold).
x̄ = (1998 + 1999 + ... + 2007) / 10 = 2003.5
ȳ = (1286 + 1307 + ... + 1452) / 10 = 1377.4
Step 2: Calculate the deviations from the means for each data point.
x - x̄: -5.5, -4.5, -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5
y - ȳ: -91.4, -70.4, -46.4, -34.4, -3.4, 18.6, 35.6, 38.6, 61.6, 75.6
Step 3: Calculate the product of the deviations for each data point.
(x - x̄)(y - ȳ): 501.70, 319.60, 162.40, 85.60, 5.10, -9.30, 17.80, 57.90, 153.40, 264.60
Step 4: Calculate the sum of the product of deviations.
Σ (x - x̄)(y - ȳ) = 501.70 + 319.60 + 162.40 + 85.60 + 5.10 - 9.30 + 17.80 + 57.90 + 153.40 + 264.60 = 1558.90
Step 5: Calculate the sum of the squared deviations for x.
Σ (x - x̄)^2 = (-5.5)^2 + (-4.5)^2 + (-3.5)^2 + (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 + (3.5)^2
= 5.5^2 + 4.5^2 + 3.5^2 + 2.5^2 + 1.5^2 + 0.5^2 + 0.5^2 + 1.5^2 + 2.5^2 + 3.5^2
= 30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25
= 92.50
Step 6: Calculate the sum of the squared deviations for y.
Σ (y - ȳ)^2 = (-91.4)^2 + (-70.4)^2 + (-46.4)^2 + (-34.4)^2 + (-3.4)^2 + (18.6)^2 + (35.6)^2 + (38.6)^2 + (61.6)^2 + (75.6)^2
= 8362.96 + 4953.60 + 2153.96 + 1183.36 + 11.56 + 345.96 + 1267.36 + 1492.96 + 3799.36 + 5700.36
= 29568.52
Step 7: Calculate the correlation coefficient r.
r = Σ (x - x̄)(y - ȳ) / sqrt( Σ (x - x̄)^2 * Σ (y - ȳ)^2 )
= 1558.90 / sqrt( 92.50 * 29568.52 )
≈ 0.997 (rounded to three decimal places)
Step 8: Find the equation of the line of best fit.
r = (∑x * ∑y - n * ∑xy) / sqrt( (∑x^2 - (1/n) * (∑x)^2) * (∑y^2 - (1/n) * (∑y)^2) )
0.997 = (10 * 2003.5 * 1377.4 - 10 * 1558.9) / sqrt( (10 * 406117.5 - (1/10) * 13951^2) * (10 * 12417371.6 - (1/10) * 19126.6^2) )
Let's solve this equation to find the equation of the line of best fit.
The equation of the line of best fit for this data set is:
y = 38.51x - 77389.11 (rounded to two decimal places)
Step 9: Predict the number of movie tickets sold in 2014.
To predict the number of movie tickets sold in 2014, we substitute x = 2014 into the equation of the line of best fit.
y = 38.51 * 2014 - 77389.11
y ≈ 77614.04
Therefore, the predicted number of movie tickets sold in 2014 is approximately 77,614 (rounded).