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Two-Variable Linear Regression

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Question 1: Calculate the following (You need to explain and provide the formula used in each one of the cases below):

Answer: The table given below shows the workings for moments, standard deviation and correlation coefficient

a) The first moment of X and Y.

Answer: The first moment is calculated by using the formula given below

E (X) = Σx/n = 3680/5 = 736

E (Y) = ΣY/n = 1380/5 = 276

b) The standard deviations of X and Y.

Answer: The standard deviation of X and Y are calculated by using the formula given below

s_x=√((Σ(x - x ¯)^2)/(n - 1))

=√(((1145 - 736)^2 + (510 - 736)^2 + (380 - 736)^2 + (530 - 736)^2 + (1115 - 736)^2)/(5 - 1))

=364.4071

s_y=√((Σ(y - y ¯)^2)/(n - 1))

=√(((465 - 276)^2 + (150 - 276)^2 + (165 - 276)^2 + (250 - 276)^2 + (350 - 276)^2)/(5 - 1))

=132.354

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c) The covariance of X and Y.

Answer: The covariance is calculated by using the formula given below

covariance= (Σ(x - x ¯ )*(y - y ¯ ))/(n - 1)

= ([(1145 - 736) + (510 - 736) + (380 - 736) + (530 - 736) + (1115 - 736) ] * [(465 - 276) + (150 - 276) + (165 - 276) + (250 - 276) + (350 - 276) ] )/(5 - 1)

= 35739

d) The correlation coefficient, called ρ, between X and Y.

Answer: The correlation coefficient is calculated by using the formula given below

r = (nΣxy - (Σx)(Σy))/(√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2))

r = (5*1194375 - 3680*1380)/(√(5*3239650 - 3680^2 ) √(5*450950 - 1380^2 ))

= 0.9263

Therefore, the required correlation coefficient is 0.9263

This indicates that there exists strong positive linear relationship between Cigarettes Consumed per capita in 1930 and Lung Cancer Deaths per million people in 1950

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Question 2: Difference in Air Pollution between the two countries.

Answer: Air pollution is considered as one of the most important and influential environmental risk to health. There is more chance for the country to reduce the risk of many diseases like heart related ailments, lung cancer, respiratory diseases (both acute and chronic) and stroke if they reduce the levels at which their air gets polluted

The differences in air pollution between the two countries might be due to

• Climatic Conditions

• Exposure Profile

• Industrial Smoke Emission

• Transportation smoke emission

• Air polluted from waste materials

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