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Homework 3 Solutions

Homework 3 Solutions - Julie Novak Tung Phan and Wei Han...

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Julie Novak, Tung Phan, and Wei Han Professor Rakhlin STAT 101 HW3 Solutions September 29th, 2011 1. Question 1 Question. #26: A fitness center weighed 53 male athletes and then measured the amount that they were able to lift. The correlation was found to be r =0.75. Can you interpret this correlation without knowing whether the weights were in pounds or kilograms or a mixture of the two? Yes, because the correlation does not depend on the scales of the two variables, but we would like to see a graph. Question. #28: After calculating the correlation, the anlayst at the fitness center in Exercise 26 discovered that the scale used to weigh the athletes was off by 5 pounds; each athlete’s weight was measured as 5 pound more than it actually was. How does this error affect the reported correlation? Mathematically, the covariance between x and y is Cov ( x, y ) = n i =1 ( x i - ¯ x )( y i - ¯ y ) n - 1 . Be- cause the original weight, (let’s say x ), was five pounds greader than the true weight, (let’s say x 0 ), we have to subtract 5 pounds from the original weight, x , to get the true weight (i.e. x 0 = x - 5). Then, the new mean of x 0 is ¯ x 0 = n X i =1 x 0 i n = n X i =1 x i - 5 n = n X i =1 x i n - 5 n = n X i =1 x i n - n X i =1 5 n = ¯ x - n * 5 n = ¯ x - 5 Also, the new standard deviation of x 0 is sd x 0 = v u u t 1 n - 1 n X i =1 ( x 0 i - ¯ x 0 ) 2 = v u u t 1 n - 1 n X i =1 ( x i - 5 - x - 5)) 2 = v u u t 1 n - 1 n X i =1 ( x i - 5 - ¯ x + 5) 2 = v u u t 1 n - 1 n X i =1 ( x i - ¯ x ) 2 = sd x 1

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Basically, the standard deviation remains the same if we subtracted five from x . Now, the new covariance, Cov ( x 0 , y ), between x 0 and y is Cov ( x 0 , y ) = n i =1 ( x 0 i - ¯ x 0 )( y i - ¯ y ) n - 1 = n i =1 (( x i - 5) - x - 5))( y i - ¯ y ) n - 1 = n i =1 (( x i - 5 - ¯ x + 5))( y i - ¯ y ) n - 1 = n i =1 ( x i - ¯ x )( y i - ¯ y ) n - 1 = Cov ( x, y ) Therefore, the new covariance Cov ( x 0 , y ) equals the original covariance Cov ( x, y ) (i.e. Cov ( x 0 , y ) = Cov ( x, y )). In addition, using the facts about the new standard deviation, sd x , we see that the new correlation is going to be the same as the original correlation. Mathematically, Corr ( x 0 , y ) = Cov ( x 0 , y ) sd x 0 sd y = Cov ( x, y ) sd x sd y = Corr ( x, y ) You can also arrive at the answer without the math. In class and in the book, corre- lation is a direct measure of the slope of the line that we fit in a scatterplot of x and y . Suppose we subtract five from x because we measured it incorrect. By doing so, we shift all the values of x to the left on the scatter plot while y still has the same values. But, when we fit the line on the scatterplot of the shifted values of x and y , the slope remains the same as the line we fit on the scatterplot with the original x and y ; only the intercept of the line changes. A visualization of this is posted below: the red points represent the weights shifted 5 pounds while the blue points are the original data. Notice how the slopes do not chang, but the intercept does.
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Homework 3 Solutions - Julie Novak Tung Phan and Wei Han...

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