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Chapter Six PSY201

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Kristie Dukewich

Chapter Six: Correlation – whether two variables are related – if related, one of them is the cause of the other – if a correlation doesn't exist between the two variables, a causal relationship can be ruled out – tests reliability of the scores – correlation and regression very much related + both involve relationship between two or more variables + correlation primarily concerned with finding out whether a relationship exists and with determining its magnitude and direction, whereas regression is primarily concerned with using the relationship for prediction Relationships: – deals with magnitude and direction of relationships Linear Relationships – graph used to plot out the relationship between X and Y variables are called scatter plot + a graph of paired X and Y values – points that fall on a straight line + when a straight line describes the relationship between two variables, it's linear + linear relationship between two variables is one in which the relationship can be most accurately represented by a straight line – some relationships are curvilinear + when a scatter plot of the X and Y variables are drawn, a curved line fits the points better than a straight line – deriving the equation of the straight line - finding the Y intercept a + Y intercept is the value of Y where the line intersects the Y axis → it's the Y value when X=0 + a = Y intercept = 500 in (0, 500) finding the slope b - slope of a line measures of its rate of change - tells how much the Y score changes for each unit change in the X score - when dealing with a straight line, the slope is constant – it doesn't matter what values we pick for X2 & X1 + corresponding Y2 & Y1 scores will yield the same value of slope - ie: if the slope is 0.40, then it means that Y value increases 0.40 unit for every 1-unit increase in X – full equation for the linear relationship that exists is + Y = bX + a + Y = 0.40X + 500 – When predicting, use the X value and place it into the formula and find the Y + could be done graphically Positive and Negative Relationships - relationship between two variables may be positive or negative - positive relationship indicates that there's a direct relationship between the variables - negative relationship indicates that there is an inverse relationship between X and Y - slope of line tells whether the relationship is positive – when it's positive, the slope is positive - the line runs upward from left to right, indicating that as X increases, Y increases → direct relationship exists between the two variables - when relationship is negative, there's an inverse relationship between the variables → making it negative. + curve runs downward from left to right + ie: as X increases, Y decreases Perfect and Imperfect Relationships - when points have fallen on the straight line, the relationship is a perfect one. + rare + a positive/a negative relationship exists and all of the points fall on the line - imperfect relationship is one in which an relationship exists, but all of the points do not fall on the line - ie: if lower values of IQ are associated with lower values of GPA and higher values of IQ are associated with higher values of GPA → perfect imperfect relationship, and linear - regression line + best-fitting line used for prediction + used cause it's an imperfect relationship → cannot draw a single straight line through all of the points Correlation: – correlation focuses on the direction and degree of the relationship + direction of the relationship refers to whether the relationship is positive/negative + degree of relationship refers to the magnitude/strength of the relationship - degree of relationship can vary from non-existent to perfect – when relationship is perfect, correlation is at its highest and can exactly predict from one variable to the other + as X changes, so does Y + the same value of X always leads to the same value of Y (works for the opposite as well) + the points all fall on a straight line → linear relationship – when relationship is non-existent, correlation is at its lowest and knowing the value of one of the variables doesn't help at all in predicting the other + imperfect relationships have intermediate levels of correlation and prediction is approximate - same value of X doesn't always lead to the same value of Y - on the average, Y changes systematically with X – can do a better job of predicting Y with knowledge of X than without it – correlation coefficient expresses quantitatively the magnitude and direction of the relationship + can vary from +1 to -1 + sign of the coefficient tells whether the relationship is positive or negative + numerical part of correlation coefficient describes the magnitude of the correlation - highers the number, greater the correlation + a correlation of +1 means correlation is perfect & relationship is positive + correlation of -1 mean correlation is perfect & relationship is negative + when relationship is non-existent, correlation coefficient equals 0 + imperfect relationships have correlation coefficients varying in magnitude between 0 and 1 – they will be plus or minus depending on the direction of the relationship – in a regression line graph, + closer the points are to the regression line, higher the magnitude of the correlation efficient and more accurate the prediction + when correlation is zero, there is no relationship between X and the Y → Y does not increase/decrease systematically with increases/decreases in X → with zero correlation, the regression line for predicting Y is horizontal and knowledge of X does not aid in predicting Y The Linear Correlation Coefficient Pearson r – ie: when shopping for a bag of oranges, want to know whether the relationship between the weight of the oranges in each bag and their cost + randomly sample six oranges, weigh each one → points fall on a straight line = there is a perfect positive correlation between the cost and weight of the oranges. +1 correlation + convert raw scores to z scores - weight (X) & cost (Y) expressed as standard scores - the paired raw scores for each bag of oranges have the same z value → all of the paired raw scores occupy the same relative position within their own distributions;; they have the same z-values. When using raw scores, relationship is obscured because of differences in scaling between the two variables. If paired scores occupy the same relative position within their own distributions, then the correlation must be perfect (r= 1) because knowing one of the paired values will allow us to exactly predict the other value. If prediction is perfect, the relationship must be perfect – Pearson r is a measure of the extent to which paired scores occupy the same or opposite positions within their own distributions + includes paired scores occupying opposite positions + if paired z scores have same magnitude but opposite signs, the correlation would again be perfect and r would equal -1 – since correlation is concerned with the relationship between variables and the variables are ofte
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