Chapter 3TOPICSLIDECorrelation Defined3Range of the Correlation Coefficient6Scatter Plots9Null and Alternative Hypotheses12Statistical Significance16Example 121Example 224Coefficient of Determination28Tutorials• Obtaining the Correlation Coefficient in Excel 2007CORRELATION AND REGRESSIONChapter 3CORRELATION➊ Indicates how well the ranking of scores on one variable matches the ranking of scores on a second variable➋ As the ranking of scores on the first variable increasingly match the ranking of scores on the second variable, the correlation will be stronger• The fewer matched rankings, the weaker the correlationChapter 3CORRELATION➌ The ranking of scores may match in the same direction (i.e., the score ranked first on variable 1 is also ranked first on variable 2) or opposite direction (i.e., the score ranked first on variable 1 is ranked last on variable 2) ➍ There is no correlation when the ranking of scores on one variable fail to match any of the scores on the second variableChapter 3CORRELATION➊ EXAMPLE: Five soccer players were ranked according to their soccer ability and their grade point average (GPA)Perfect Positive rPerfect Negative rSoccer SoccerPlayer Ability GPA Player Ability GPAA 1 1 A 1 5B 2 2 B 2 4C 3 3 C 3 3D 4 4 D 4 2E 5 5 E 5 1Chapter 3CORRELATION➊ The numeric value of the correlation coefficient has a range of +1.00 to -1.00, where zero indicates no correlation• The closer the correlation coefficient is to +1.00 or -1.00, the stronger the correlation between two variables• The closer the correlation coefficient is to 0, the weaker the correlation between two variables• A correlation coefficient equal to 0 means there is no correlation between two variables➋ Which value represents a stronger correlation?• +.65 or -.85Chapter 3CORRELATION➊ The correlation coefficient describes two characteristics:• The sign of the correlation (positive or negative) indicates the direction of the relationship between the two variables• The value of the correlation indicates how strong the correlation is between two variables➋ The symbol for the correlation between two variables for a sample is a lower case, italicized rChapter 3CORRELATION➊ Here is a rough guideline for defining the strength of a correlation coefficient:• r = ±.80 to ±1.00  Strong Correlation• r = ±.60 to ±.80  Moderate Correlation• r = ±.40 to ±.60  Weak to Moderate • r < ±.40  Weak Correlation➋ The guideline above assumes a sample size of N ≥ 30Chapter 3CORRELATION➊ A scatter plot is a graph that describes the direction and strength of the correlation between two variables➋ The closer the points in the graph are to forming a straight line, the stronger the correlation between the two variables• When the points in the graph form a circular pattern, the correlation will be close or equal to zero• When the pattern of points leans from lower right to upper left, the scatter plot indicates the correlation is negative• When the pattern of points leans from lower left to upper right, the scatter plot indicates the correlation is positiveChapter 3SCATTER PLOTS➊ When the pattern is lower right to upper left, the correlation is negative:➋ When the pattern is lower left to upper right, the correlation is positive:Chapter 3SCATTER PLOTSYXYXChapter 3Scatter Plots➊ A non-zero correlation does not necessarily mean two variables are related to each other➋ There are two competing hypotheses:• The alternative hypothesis (HA) contends there is a true correlation between the two variables for the population and the sample correlation observed is not solely due to random error• The null hypothesis (H0) states that there is no correlation between the two variables for the population and that any sample correlation observed is solely due to random errorChapter 3NULL HYPOTHESIS➊ When a correlation coefficient is sufficiently large, we can make the inference that it reflects not just random error alone, but also a measure of how much two variables have in common• Remember random error is present in everything we measure – you can’t get rid of it and all statistics contain some amount of random error• Smaller samples have more random error and larger samples have lessChapter 3NULL HYPOTHESIS➊ A statistical conclusion is a statement that rejects or fails to reject the null hypothesis• When we reject the null hypothesis, we are saying the sample correlation obtained is NOT solely due to random error but indicates a real correlation between the two variables for the population• When we fail to reject the null hypothesis, we are acknowledging the observed sample correlation may be only due to random error and that there may not be any true correlation between the two variables for the populationChapter 3NULL HYPOTHESIS➊ The stronger the correlation, the more likely there is a real correlation between two variables for the population➋ Whether a sample correlation between two variables is real or not is a function of how big the sample size is and the strength of the correlation between two variables• As a general rule, the larger the sample size, the weaker the sample correlation needs to be in order to declare it statistically significant (meaning the null hypothesis is rejected)• In other words, the correlation coefficient needs to be increasingly stronger for data sets based on small sample sizesChapter 3NULL HYPOTHESIS➊ To determine if a sample correlation is significant, we need to first work from the assumption that the null hypothesis is true • We assume the null hypothesis is true because we haven’t analyzed the data yet (there’s no evidence of a correlation without analyzing the data)➋ We only analyze the data from one sample, but to determine if a sample correlation is statistically significant we have to remember there are an infinite number of samples that could have been selectedChapter 3STATISTICAL SIGNIFICANCE➊ Assuming the null hypothesis is true, the correlation for the sample obtained should be zero and if the value is not zero, then we assume the correlation is solely due to random error➋ If we imagine obtaining the correlations for all possible samples (where each sample is the same size), we would find that the average of all sample correlations is equal to the population correlation• Again, if the null hypothesis is true, the correlation between two variables for the population will be