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QuestionCalculate the correlation coefficient between annual growth of National Income and Gross Domestic Saving from 1992-2002.

Studdy Solution

STEP 1

Assumptions1. The annual growth of National Income and Gross Domestic Saving as a percentage of GDP are two numerical variables. . We are asked to find the correlation coefficient, which measures the strength and direction of the linear relationship between these two variables.
3. We have10 pairs of data points.

STEP 2

First, we need to calculate the mean of each variable. The mean is calculated by summing all the values and dividing by the number of values.
For the annual growth of National Incomexˉ=∑xn\bar{x} = \frac{\sum x}{n}For Gross Domestic Saving as a percentage of GDPyˉ=∑yn\bar{y} = \frac{\sum y}{n}

STEP 3

Calculate the mean of the annual growth of National Income.
xˉ=14+17+18+17+16+12+16+11+8+1010\bar{x} = \frac{14 +17 +18 +17 +16 +12 +16 +11 +8 +10}{10}

STEP 4

Calculate the mean of Gross Domestic Saving as a percentage of GDP.
yˉ=24+23+26+27+25+25+23+25+24+2310\bar{y} = \frac{24 +23 +26 +27 +25 +25 +23 +25 +24 +23}{10}

STEP 5

Next, we need to calculate the standard deviation of each variable. The standard deviation is a measure of the amount of variation or dispersion of a set of values.
For the annual growth of National Incomesx=∑(x−xˉ)2n−1s_x = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}For Gross Domestic Saving as a percentage of GDPsy=∑(y−yˉ)2n−1s_y = \sqrt{\frac{\sum (y - \bar{y})^2}{n-1}}

STEP 6

Calculate the standard deviation of the annual growth of National Income.
sx=(14−xˉ)2+(17−xˉ)2+(18−xˉ)2+(17−xˉ)2+(16−xˉ)2+(12−xˉ)2+(16−xˉ)2+(11−xˉ)2+(8−xˉ)2+(10−xˉ)210−1s_x = \sqrt{\frac{(14 - \bar{x})^2 + (17 - \bar{x})^2 + (18 - \bar{x})^2 + (17 - \bar{x})^2 + (16 - \bar{x})^2 + (12 - \bar{x})^2 + (16 - \bar{x})^2 + (11 - \bar{x})^2 + (8 - \bar{x})^2 + (10 - \bar{x})^2}{10-1}}

STEP 7

Calculate the standard deviation of Gross Domestic Saving as a percentage of GDP.
sy=(24−yˉ)2+(23−yˉ)2+(26−yˉ)2+(27−yˉ)2+(25−yˉ)2+(25−yˉ)2+(23−yˉ)2+(25−yˉ)2+(24−yˉ)2+(23−yˉ)210−1s_y = \sqrt{\frac{(24 - \bar{y})^2 + (23 - \bar{y})^2 + (26 - \bar{y})^2 + (27 - \bar{y})^2 + (25 - \bar{y})^2 + (25 - \bar{y})^2 + (23 - \bar{y})^2 + (25 - \bar{y})^2 + (24 - \bar{y})^2 + (23 - \bar{y})^2}{10-1}}

STEP 8

Finally, we can calculate the correlation coefficient. The correlation coefficient is calculated using the following formular=∑(x−xˉ)(y−yˉ)(n−1)sxsyr = \frac{\sum (x - \bar{x})(y - \bar{y})}{(n-1)s_x s_y}

STEP 9

Calculate the correlation coefficient.
r=(14−xˉ)(24−yˉ)+(17−xˉ)(23−yˉ)+(18−xˉ)(26−yˉ)+(17−xˉ)(27−yˉ)+(16−xˉ)(25−yˉ)+(12−xˉ)(25−yˉ)+(16−xˉ)(23−yˉ)+(11−xˉ)(25−yˉ)+(8−xˉ)(24−yˉ)+(−xˉ)(23−yˉ)(−)sxsyr = \frac{(14 - \bar{x})(24 - \bar{y}) + (17 - \bar{x})(23 - \bar{y}) + (18 - \bar{x})(26 - \bar{y}) + (17 - \bar{x})(27 - \bar{y}) + (16 - \bar{x})(25 - \bar{y}) + (12 - \bar{x})(25 - \bar{y}) + (16 - \bar{x})(23 - \bar{y}) + (11 - \bar{x})(25 - \bar{y}) + (8 - \bar{x})(24 - \bar{y}) + ( - \bar{x})(23 - \bar{y})}{(-)s_x s_y}The correlation coefficient is the solution to the problem. The value of the correlation coefficient will be between - and. A value of indicates a perfect positive correlation, a value of - indicates a perfect negative correlation, and a value of indicates no correlation.

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