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 value of the correlation coefficient (r), we first need to find the mean for both the years and the number of movie tickets sold.
Year mean (x̄):
1998 + 1999 + 2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006 + 2007 = 20,085
x̄ = 20,085 / 10 = 2,008.5
Tickets Sold mean (ȳ):
1286 + 1307 + 1331 + 1343 + 1374 + 1396 + 1412 + 1415 + 1438 + 1452 = 13,054
ȳ = 13,054 / 10 = 1,305.4
Next, we calculate the following values to find r:
Σ((x - x̄) * (y - ȳ)): The sum of the product of the differences between the years and the mean of years, and the differences between the number of tickets sold and the mean of tickets sold.
Σ((x - x̄)^2): The sum of the squared differences between the years and the mean of years.
Σ((y - ȳ)^2): The sum of the squared differences between the number of tickets sold and the mean of tickets sold.
Using the given data:
Σ((x - x̄) * (y - ȳ)) = (1998 - 2,008.5) * (1286 - 1,305.4) + (1999 - 2,008.5) * (1307 - 1,305.4) + ... + (2007 - 2,008.5) * (1452 - 1,305.4)
= -10.5 * -19.4 + -9.5 * 1.6 + ... + -1.5 * 146.6
= 513.7
Σ((x - x̄)^2) = (1998 - 2,008.5)^2 + (1999 - 2,008.5)^2 + ... + (2007 - 2,008.5)^2
= (-10.5)^2 + (-9.5)^2 + ... + (-1.5)^2
= 1690
Σ((y - ȳ)^2) = (1286 - 1,305.4)^2 + (1307 - 1,305.4)^2 + ... + (1452 - 1,305.4)^2
= (-19.4)^2 + (1.6)^2 + ... + (146.6)^2
= 392,086.56
Using these values, we can calculate the correlation coefficient (r):
r = Σ((x - x̄) * (y - ȳ)) / √(Σ((x - x̄)^2) * Σ((y - ȳ)^2))
= 513.7 / √(1690 * 392,086.56)
= 513.7 / √(663,761,945.6)
≈ 513.7 / 25764.54
≈ 0.0199
The correlation coefficient (r) is approximately 0.0199.
Now, to predict the number of movie tickets sold in 2014, we can use the linear regression equation:
y = mx + b
Where:
m = r * (Sy / Sx)
b = ȳ - (m * x̄)
x = 2014 (the year for which we want to predict the number of tickets sold)
y = predicted number of movie tickets sold in 2014
Sy = √(Σ((y - ȳ)^2) / (n - 1)) (Standard deviation of y-values)
Sx = √(Σ((x - x̄)^2) / (n - 1)) (Standard deviation of x-values)
n = number of data points (10 in this case)
Using the calculated values:
Sy = √(Σ((y - ȳ)^2) / (n - 1)) = √(392,086.56 / (10 - 1)) = √(392,086.56 / 9) ≈ √43,565.173 ≈ 208.8818
Sx = √(Σ((x - x̄)^2) / (n - 1)) = √(1690 / (10 - 1)) = √(1690 / 9) ≈ √187.7777 ≈ 13.6902
Now, substituting the values into the linear regression equation:
m = r * (Sy / Sx) ≈ 0.0199 * (208.8818 / 13.6902) ≈ 0.3049
b = ȳ - (m * x̄) = 1,305.4 - (0.3049 * 2,008.5) ≈ 1,305.4 - 613.2327 ≈ 692.1673
For x = 2014, the predicted number of movie tickets sold (y) is:
y = mx + b = 0.3049 * 2014 + 692.1673 = 612.5266 + 692.1673 = 1304.6939
Therefore, the predicted number of movie tickets sold in 2014 is approximately 1,305 (rounded to the nearest whole number).