# CU-Boulder MCEN 3037 - 12R (15 pages)

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## 12R

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## 12R

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Mcen 3037 - Data Analysis
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Basic Idea Vary the independent variable x Vary the dependent variable y We are trying to find out Does y depend on x Sometimes this is clear Other times it is not Is there a trend here Probably We need a statistical parameter to determine if there is a trend n rxy 1 r xy x x y i 1 i i mean x y 1 2 2 2 x x y y i i i 1 i 1 n n mean y 1 But how do we interpret this rxy 1 Indicates a perfect linear relationship with a positive slope rxy 1 Indicates a perfect linear relationship with a negative slope rxy 0 Indicates no linear correlation between x and y Unchanged by multiplying each value of a variable by a constant Unchanged by adding a constant to each variable Unchanged by interchanging x and y For a given sample size we need to determine whether a given rxy is significant or a result of pure chance For practical problems we can consult a table General procedure Calculate rxy from the measured data Determine level of significance a required a gives the probability that an experimental value of rxy will be greater by pure chance a 0 05 rt value at which 5 chance due to random effects Compare rt to rxy if rxy rt then confidence level is confirmed 95 confidence often used in engineering Say we have a sample size of 18 and we get a correlation coefficient of 0 49 Does a linear relationship exist between our variables 95 Yes our correlation coefficient is greater than that required for 95 Confidence see table It s thought that the lap times for a race car depend on the ambient temperature The following data for the same car with the same driver were measured at different races Ambient temperature F 40 47 55 62 66 88 Lap time s 65 3 66 5 67 3 67 8 67 66 6 Does a linear relationship exist between the two variables 68 Lap Time s Let s look at the plot 67 66 65 40 50 60 70 80 Temperature F 90 Ambient temperature F 40 47 55 62 66 88 Lap time s 65 3 66 5 67 3 67 8 67 66 6 Now we calculate the correlation coefficient n rxy x x y i 1 i i y 1 2 2 2 x x y y i i i 1 i 1 n n rxy 0

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