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A scatterplot is a type of graph used in statistics to display the relationship between two numerical variables. In this particular problem, we are given two sets of data: the 'x' values and the 'y' values. To create a scatterplot, plot each pair of values \((x, y)\) onto the Cartesian coordinate system. For example, the first pair is (10, 7.46). Each dot on the scatterplot represents a pair of values.
Scatterplots are extremely useful because they visually show you how one variable might affect another. You can observe patterns, trends, and possible outliers that could influence your data analysis.
For example, in our dataset:
\( (10, 7.46), (8, 6.77), (13, 12.74), (9, 7.11), (11, 7.81), (14, 8.84), (6, 6.08), (4, 5.39), (12, 8.15), (7, 6.42), (5, 5.73) \).
Plotting these points will show you the overall trend and any deviations from a pattern, which may be crucial for understanding the underlying relationship.

The linear correlation coefficient, represented as \((r)\), measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1.
To calculate \((r)\) for our dataset, we use the formula:
\[ r = \frac{n(\text{sum of } xy) - (\text{sum of } x)(\text{sum of } y) }{ \sqrt{ [n \text{sum of } x^2 - (\text{sum of } x)^2][n \text{sum of } y^2 - (\text{sum of } y)^2] } } \]
First, calculate the necessary summations:
\(\text{sum of } x = 99\), \(\text{sum of } y = 82.5\),
\(\text{sum of } xy = 915.16\), \(\text{sum of } x^2 = 1105\), and \(\text{sum of } y^2 = 761.86\)
Substitute these into the formula to obtain:
\[ r = \frac{11(915.16) - (99)(82.5)}{\sqrt{[11(1105) - (99)^2][11(761.86) - (82.5)^2]}} = 0.8164. \]
A value of 0.8164 suggests a strong positive linear relationship between the variables in our dataset. To formally evaluate this, compare it against a critical value at a chosen significance level, typically 0.05. For our 11 data pairs, 0.8164 is higher than the critical value, thus supporting a significant linear correlation.

The sum of squares (SS) represents the total squared deviations from the mean of a dataset. In our calculations, we use it to find both the variances and the correlation coefficient.
We compute different types of sums of squares:
- Sum of squares of x (\text{SS}_x) and y (\text{SS}_y): These give us the variability of x and y values, respectively. \[ \text{SS}_x = n \sum x^2 - (\sum x)^2\]
\[ \text{SS}_y = n \sum y^2 - (\sum y)^2\]
Using our dataset:
\[ \text{SS}_x = 11 \times 1105 - 99^2 = 11255 - 9801 = 1454\text{ SS}_x = 1454 \]
\[ \text{SS}_y = 11 \times 761.86 - 82.5^2 = 8380.46 - 6806.25 = 1574.21\ \text{SS}_y = 1574.21 \]
- Sum of squares of the product xy (\text{SS}_{xy}):
\[ \text{SS}_{xy} = n\sum (xy) - (\sum x)(\sum y)\]
\[ \text{SS}_{xy} = 11 \times 915.16 - 99 \times 82.5 = 10066.76 - 8167.5 = 1899.26 \text{ SS}_{xy} = 1899.26 \]
These sums are essential to substitute into various formulas, like the linear correlation coefficient. They offer insight into the underlying structure of the data and its variance. Understanding these components deeply enhances statistical analysis, ensuring well-supported conclusions.