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# Correlation Coefficient In Minitab Paper

In regression analysis, we have,

Minimized the slope and the intercept

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Correlation Coefficient In Minitab Paper
Just from $9/Page Predicted y as y hat in raw units It tells about which line of regression fits the data best but does not tell about the relative strength of the relationship image If a simple linear regression model is considered, {y_i} = {\beta _0} + {\beta _1}{x_i} + {e_i}y i â€‹ =Î² 0 â€‹ +Î² 1 â€‹ x i â€‹ +e i â€‹ . Where, \beta _0:Î² 0 â€‹ : intercept of the regression equation or shift from the origin. \beta_1:Î² 1 â€‹ :slope of the equation. e_i:e i â€‹ :random error and {e_i}\sim iid\ N\left({0,{\sigma ^2}} \right)e i â€‹ âˆ¼iid N(0,Ïƒ 2 ). Then, {\beta _1} = \frac{{{\mathop{\rm cov}} (x,y)}}{{S_x^2}}Î² 1 â€‹ = S x 2 â€‹ cov(x,y) â€‹ , where, {\mathop{\rm cov}} (x,y) = {S_{xy}} = \sum\limits_i {{x_i}} {y_i} – n\bar x\bar y \ , \ \bar X = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} ,\ \, \bar Y = \frac{1}{n}\sum\limits_{i = 1}^n {{y_i}}cov(x,y)=S xy â€‹ = i âˆ‘ â€‹ x i â€‹ y i â€‹ âˆ’n x Ë‰ y Ë‰ â€‹ , X Ë‰ = n 1 â€‹ i=1 âˆ‘ n â€‹ x i â€‹ , Y Ë‰ = n 1 â€‹ i=1 âˆ‘ n â€‹ y i â€‹ . S_{xy}S xy â€‹ can be written as S_{xy} = \sum\limits_{i = 1}^n {\left( {{x_i} – \bar x} \right)} \left( {{y_i} – \bar y} \right)S xy â€‹ = i=1 âˆ‘ n â€‹ (x i â€‹ âˆ’ x Ë‰ )(y i â€‹ âˆ’ y Ë‰ â€‹ ). Now, standardize X and Y by subtracting their means and dividing by their standard deviation. It becomes the Pearsonâ€™s Product moment correlation as follows, r = \frac{{\sum\limits_{i = 1}^n {\left( {{x_i} – \bar x} \right)} \left( {{y_i} – \bar y} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} – \bar x} \right)}^2}} \sum\limits_{i = 1}^n {{{\left( {{y_i} – \bar y} \right)}^2}} } }}r= i=1 âˆ‘ n â€‹ (x i â€‹ âˆ’ x Ë‰ ) 2 i=1 âˆ‘ n â€‹ (y i â€‹ âˆ’ y Ë‰ â€‹ ) 2 â€‹ i=1 âˆ‘ n â€‹ (x i â€‹ âˆ’ x Ë‰ )(y i â€‹ âˆ’ y Ë‰ â€‹ ) â€‹ Correlation can be computed between the following variables: Weight and height Blood pressure and weight Wheat yield and temperature Food habits of a mother and the health of a baby Age of a house and the price of a house Time spent in the gym and the health of a person etc. A scatter diagram is the easiest way to visualize correlation between 2 variables. It shows the values of two variables, X and Y graphically, and how both variables are related to each other. 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