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Confidence Intervals and Hypothesis Testing with Correlation Coefficients

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Confidence Intervals and Hypothesis Testing with Correlation Coefficients

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    1. Confidence Intervals and Hypothesis Testing with Correlation Coefficients

    2. Inspecting Scatterplots :The Bivariate Normal Distribution

    3. Inspecting Scatterplots :The Bivariate Normal Distribution

    4. Inspecting Scatterplots :The Bivariate Normal Distribution

    5. Confidence Intervals for Correlation Coefficients

    6. Confidence Intervals for Correlation Coefficients Biased and Unbiased estimators

    7. Confidence Intervals for Correlation Coefficients r is a sample statistic and r is a population parameter that represents the correlation between two variables (X and Y) in the population. Confidence intervals for correlations must be computed differently from confidence intervals about means (or mean differences) because the sampling distribution of r is skewed, particularly as r approaches 1 and -1.

    8. Confidence Intervals for Correlation Coefficients To deal with this problem we use the Fisher r-to-Zr transformation. Formally Zr = 0.5 loge [(1 + r)/(1 - r)] or, you can look it up in tables (Handout) or, use the function Fisher(r) in Excel to find Zr and FisherInv(Zr ) to find r. The advantage of Zr is that its sampling distribution is normal.

    9. Confidence Intervals for Correlation Coefficients We next ask: what is the standard error of Zr? SEZr = or Because Zr is normally distributed and its standard error(SEZr) is defined, we can place a 95% confidence interval around Zr as follows CI = Zr 1.96 (SEZr ) The limits of the CI can then be converted back to rs using FisherInv (Zr) in Excel.

    10. Confidence Intervals for Correlation Coefficients

    11. Testing whether r is different from 0 When r = 0 and the sample size is relatively large, the sampling distribution of r will be normal with a standard error of which can be estimated by Therefore, we can calculate a t-statistic for a correlation coefficient as:

    12. Testing whether r is different from 0

    13. Testing whether r is different from 0

    14. Testing whether r is different from 0

    15. Testing whether r is different from 0

    16. Testing whether r is different from a known r When r ? 0 the sampling distribution of r will not be normal (in general) so the Fisher transform is used. Non-directional test (that r = .5, a = .05)

    17. Testing whether r is different from a known r When r ? 0 and the sampling distribution of r will not be normal (in general) so the Fisher transform is used. Directional test (that r > .5, a = .05)

    18. Testing whether r is different from a known r

    19. Testing whether r is different from a known r

    20. Testing whether two independent correlations differ from each other Again, r is converted to Zr and the z-distribution is consulted

    21. Testing whether two independent correlations differ from each other

    22. Testing whether two independent correlations differ from each other

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