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).
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.
1 answer