1. Probabilistic and Statistical Techniques 1 Lecture 24 Eng. Ismail Zakaria El Daour 2010

2. Correlation Probabilistic and Statistical Techniques

3. Overview This chapter introduces important methods for making inferences about a correlation (or relationship) between two variables, and describing such a relationship with an equation that can be used for predicting the value of one variable given the value of the other variable. Probabilistic and Statistical Techniques

4. Key Concept This section introduces the linear correlation coefficient r, which is a numerical measure of the strength of the relationship between two variables representing quantitative data. Because technology can be used to find the value of r, it is important to focus on the concepts in this section, without becoming overly involved with tedious arithmetic calculations. Probabilistic and Statistical Techniques

5. Definition one of them is related to the other A correlation exists between two variables when one of them is related to the other in some way. Probabilistic and Statistical Techniques

6. Definition r The linear correlation coefficient r measures the strength of the linear relationship between paired x- and y- quantitative values in a sample. Probabilistic and Statistical Techniques

7. Scatter plots of Paired Data Probabilistic and Statistical Techniques

8. Scatter plots of Paired Data Probabilistic and Statistical Techniques

9. Requirements 1.The sample of paired ( x, y ) data is a random sample of independent quantitative data. 2.Visual examination of the scatter plot must confirm that the points approximate a straight- line pattern. 3.The outliers must be removed if they are known to be errors. The effects of any other outliers should be considered by calculating r with and without the outliers included. Probabilistic and Statistical Techniques

10. Notation for the Linear Correlation Coefficient n represents the number of pairs of data present.  denotes the addition of the items indicated.  x denotes the sum of all x- values.  x 2 indicates that each x- value should be squared and then those squares added. (  x ) 2 indicates that the x- values should be added and the total then squared.  xy indicates that each x -value should be first multiplied by its corresponding y- value. After obtaining all such products, find their sum. r represents linear correlation coefficient for a sample.  represents linear correlation coefficient for a population.

11. r The linear correlation coefficient r measures the strength of a linear relationship between the paired values in a sample. Formula Probabilistic and Statistical Techniques

12. r Example: Calculating r Using the simple random sample of data below, find the value of r. Probabilistic and Statistical Techniques

13. Example: Calculating r - cont Probabilistic and Statistical Techniques

14. Example: Calculating r - cont Probabilistic and Statistical Techniques

15. Properties of the Linear Correlation Coefficient r –1  r  1 1. –1  r  1 r 2. The value of r does not change if all values of either variable are converted to a different scale. 3. The value of r is not affected by the choice of x and y. Interchange all x- and y- values and the value of r will not change. 4. r measures strength of a linear relationship. Probabilistic and Statistical Techniques

16. Explained Variation (coefficient of variation) Explained Variation (coefficient of variation) r 2 y The value of r 2 is the proportion of the variation (coefficient of variation) in y that is explained by the linear relationship between x and y. Probabilistic and Statistical Techniques

17. r = 0.926 Using the duration/interval data an Example, we have found that the value of the linear correlation coefficient r = 0.926. What proportion of the variation? With r = 0.926, we get r 2 = 0.857. (y) x and y We conclude that 0.857 (or about 86%) of the variation (y) can be explained by the linear relationship between x and y. y x and y This implies that 14% of the variation in y after cannot be explained by the linear relationship between x and y. Example: Probabilistic and Statistical Techniques

18. Common Errors Involving Correlation 1. Averages: Averages suppress individual variation and may inflate the correlation coefficient. 2. Linearity: There may be some relationship between x and y even when there is no linear correlation. Probabilistic and Statistical Techniques

19. Note: confidence interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. Probabilistic and Statistical Techniques

20. Confidence Interval for Population Proportion where Probabilistic and Statistical Techniques

21. General Example  The figure reveals that μ lies in the 95.44% confidence interval in 19 of the 20 samples, that is, in 95% of the samples  We can be 95.44% confident that any computed 95.44% confidence interval will contain μ. Probabilistic and Statistical Techniques

22. Example The U.S. Bureau of Labor Statistics collects information on the ages of people in the civilian labor force and publishes the results in Employment and Earnings. Fifty people in the civilian labor force are randomly selected. Find a 95% confidence interval for the mean age, μ =36.38, of all people in the civilian labor force. Assume that the population standard deviation of the ages is 12.1 years. Probabilistic and Statistical Techniques

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