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Correlation

Correlation. Review. We saw in the past that when you have two numeric or quantitative variables one thing to explore is the correlation between the two variables. The basic idea is to see if as one variable changes is there a systematic change in the other variable.

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Correlation

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  1. Correlation

  2. Review We saw in the past that when you have two numeric or quantitative variables one thing to explore is the correlation between the two variables. The basic idea is to see if as one variable changes is there a systematic change in the other variable. The correlation coefficient is a number that is used as a summary for the relationship. The correlation coefficient, by the way it is constructed, can have any value from a -1 to 1. -1 0 1

  3. Review A correlation coefficient of 0 is an indication that there is no relationship between the two variables. The closer the coefficient is to -1 or 1 the stronger is the relationship between the two variable. On the minus side the variables move in opposite directions and are said to be negatively correlated. On the plus side the variables move in the same direction and said to be positively correlated. Examples We see from economics that the relationship between the interest rate and the level of investment in plant and equipment is negatively related. From advertising the thinking is that modest increases in the advertising budget lead to greater sales – a positive relation.

  4. Inference If you are interested in a relationship between two variables in the population, but only take a sample, then you can use hypothesis testing as a way to think about the population. The Greek small letter rho, ρ, is used as the symbol for the population correlation coefficient. Under the null hypothesis it is assumed that rho = 0 and the alternative we will explore here is a two tailed alternative of rho not equal to 0. IN symbols we have Ho: ρ = 0 H1: ρ≠ 0.

  5. T test The sample correlation coefficient r will be used as a basis for the test and in a repeated sampling context r has a t distribution with n – 2 degrees of freedom. Under the critical value approach you look up in a t table for the critical values of t based on the degrees of freedom and the alpha value for the test. Since we have a two tailed test you split alpha in half. Then if the t stat based on the sample is more extreme (or farther away from 0) than the critical values you reject Ho and go with the alternative. The is the same as saying there is a relationship between the two variables.

  6. T test Under the p-value approach to the hypothesis test you calculate the t stat based on the sample information. Then you find the tail area for the stat and multiply it by 2 to get the p-value because we have a two tailed test. If the p-value is less than the alpha for the test then the null hypothesis can be rejected and we would go with the alternative hypothesis.

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