LaTasha was presented with the following data set and argued that there was no correlation between [tex]$x$[/tex] and [tex]$y$[/tex]. Is LaTasha correct? Use the regression equation to explain your reasoning.
| [tex]$x$[/tex] | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| [tex]$y$[/tex] | 4 | 5 | 4 | 5 | 4 | 5 | 4 |
Answer (2)
Calculate the mean of x ( x ˉ ) and y ( y ˉ ). x ˉ = 4 and y ˉ = 7 31 .
Calculate the Pearson correlation coefficient (r). r = 0 .
Determine the regression equation y = a + b x , where b = r ( s x s y ) and a = y ˉ − b x ˉ . Since r = 0 , b = 0 and a = 7 31 .
Conclude that there is no correlation between x and y because r = 0 , and the regression equation is y = 7 31 , which means y does not depend on x. LaTasha is correct. T r u e .
Explanation
Problem Analysis We are given a data set with x and y values and asked to determine if there is a correlation between them. LaTasha claims there is no correlation. We will use the regression equation to investigate this claim.
Calculate Means First, we need to calculate the means of the x and y values. The x values are 1, 2, 3, 4, 5, 6, and 7. The y values are 4, 5, 4, 5, 4, 5, and 4.
Mean of x The mean of the x values, denoted as x ˉ , is calculated as: x ˉ = 7 1 + 2 + 3 + 4 + 5 + 6 + 7 = 7 28 = 4
Mean of y The mean of the y values, denoted as y ˉ , is calculated as: y ˉ = 7 4 + 5 + 4 + 5 + 4 + 5 + 4 = 7 31 ≈ 4.4286
Calculate Standard Deviations Next, we calculate the standard deviations of the x and y values. The standard deviation of the x values, denoted as s x , is 2. The standard deviation of the y values, denoted as s y , is approximately 0.4949.
Pearson Correlation Coefficient Now, we calculate the Pearson correlation coefficient, r , using the formula: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where n is the number of data points.
Calculating r Using the values we have: r = ( 7 − 1 ) ( 2 ) ( 49 6 ) ( 1 − 4 ) ( 4 − 7 31 ) + ( 2 − 4 ) ( 5 − 7 31 ) + ( 3 − 4 ) ( 4 − 7 31 ) + ( 4 − 4 ) ( 5 − 7 31 ) + ( 5 − 4 ) ( 4 − 7 31 ) + ( 6 − 4 ) ( 5 − 7 31 ) + ( 7 − 4 ) ( 4 − 7 31 ) r = 6 ⋅ 2 ⋅ 7 6 ( − 3 ) ( − 7 3 ) + ( − 2 ) ( 7 4 ) + ( − 1 ) ( − 7 3 ) + ( 0 ) ( 7 4 ) + ( 1 ) ( − 7 3 ) + ( 2 ) ( 7 4 ) + ( 3 ) ( − 7 3 ) r = 7 12 6 7 9 − 7 8 + 7 3 + 0 − 7 3 + 7 8 − 7 9 r = 7 12 6 0 = 0
Regression Equation The regression equation is of the form y = a + b x , where b = r ( s x s y ) and a = y ˉ − b x ˉ .
Calculating a and b Since r = 0 , we have b = 0 ( s x s y ) = 0 . Then, a = y ˉ − b x ˉ = 7 31 − 0 ( 4 ) = 7 31 ≈ 4.4286 .
Regression Equation Result Therefore, the regression equation is y = 7 31 + 0 x = 7 31 . This means that the predicted value of y is always 7 31 , regardless of the value of x .
Conclusion Since the correlation coefficient r = 0 , there is no linear correlation between x and y . The regression equation confirms this, as the predicted value of y does not depend on x . Therefore, LaTasha is correct in arguing that there is no correlation between x and y .
Examples
Understanding correlation and regression is very useful in analyzing real-world data. For example, if you are tracking the number of hours you study ( x ) and your exam scores ( y ), you can use correlation to see if there's a relationship between them. If the correlation is high, it suggests that more study hours lead to higher scores. The regression equation can then help you predict your exam score based on the number of hours you study. This kind of analysis is used in many fields, from education to economics, to understand relationships between different variables and make predictions.
LaTasha is correct in stating there is no correlation between x and y . The Pearson correlation coefficient r = 0 , indicating no linear relationship, and the regression equation y = 7 31 shows that y remains constant regardless of x .
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