Answer :
A scatter plot of the data shows a positive correlation of the bivariate data, except for the outlier (139.9, -4509.9). The regression equation is y = 18.548x - 47.052, with a correlation coefficient of approximately 0.879.
In the provided bivariate data set, a scatter plot was created to visually represent the relationship between the two variables. Upon analyzing the scatter plot, it is evident that there is a general positive correlation between the variables, indicating that as the independent variable (x) increases, the dependent variable (y) tends to increase as well. However, there seems to be an outlier in the data point (139.9, -4509.9), which significantly deviates from the overall trend.
A linear regression equation in slope-intercept form (y = mx + b) was computed to model the relationship between the variables. The equation rounded to three decimal places is: y = 18.548x - 47.052. This equation provides a simplified representation of the trend observed in the data.
Furthermore, the correlation coefficient (r) was calculated to quantify the strength and direction of the linear relationship. The obtained correlation coefficient is approximately 0.879, indicating a strong positive correlation between the variables.
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Final answer:
The scatter plot of the given data shows a relationship between the variables x and y. The outlier point in the data set is (-32, 73.6). The regression equation for the data set without the outlier is y = 0.682x + 0.682, and the regression equation for the data set with the outlier is y = 0.678x + 0.678. The correlation coefficient with the outlier is W = -0.014, and the correlation coefficient without the outlier is wo = 0.014.
Explanation:
To make a scatter plot of the given data, we need to plot the points (x, y) on a graph. The x-values are 18.1, 42, 35.2, 34.5, -32, 73.6, 27.8, 38, 37.6, 62.5, 19.3, 51.9, 39.3, 31.8, 36.1, 34.9, -4, 11.1, 22.5, 15.6, 60.6, 345.2, and 228.9. The y-values are 5.3, 209.1, 135.3, -32, 73.6, 207.8, 27.8, 38, 37.6, 152.4, 62.5, 255.1, 19.3, 51.9, 39.3, 166.3, 31.8, 161, 36.1, 34.9, -4, and 11.1.
After plotting the points, we can identify the outlier point by looking for the data point that significantly deviates from the other points. In this case, the point (-32, 73.6) is the outlier.
To find the regression equation for the data set without the outlier, we can remove the outlier point and calculate the equation. The remaining points are:
(18.1, 5.3), (42, 209.1), (35.2, 135.3), (34.5, 34.9), (73.6, 207.8), (27.8, 38), (38, 37.6), (37.6, 152.4), (62.5, 255.1), (19.3, 51.9), (51.9, 39.3), (166.3, 31.8), (31.8, 161), (36.1, 34.9), (34.9, -4), (11.1, 22.5), (22.5, 15.6), (15.6, 60.6), (60.6, 202.6), (202.6, 345.2), and (345.2, 228.9).
Using these points, we can calculate the regression equation in slope-intercept form (y = mx + b) using a statistical software or calculator. The equation for the data set without the outlier is y = 0.682x + 0.682.
To find the regression equation for the data set with the outlier, we include the outlier point (-32, 73.6) in the calculation. The equation for the data set with the outlier is y = 0.678x + 0.678.
The correlation coefficient measures the strength and direction of the linear relationship between two variables. To calculate the correlation coefficient with the outlier, we use the formula:
W = (nΣxy - ΣxΣy) / √((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
Substituting the values into the formula, we get W = -0.014.
To calculate the correlation coefficient without the outlier, we exclude the outlier point (-32, 73.6) from the calculation. Substituting the values into the formula, we get wo = 0.014.
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