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A scatter diagram is a visual representation used to identify the relationship between two variables. It is a simple yet powerful tool in statistics to observe trends or patterns that might exist between the variables. In this exercise, the two variables in question are \(x\) and \(y\). By plotting each pair of \((x_i, y_i)\) on a Cartesian plane, we lay out the positioning of these variable pairs.

The horizontal axis (often the \(x\)-axis) represents the variable \(x\), while the vertical axis (the \(y\)-axis) represents\(y\). Each plotted point corresponds to a pair of \((x, y)\) values. For instance, if you plot the pairs: (4, 50), (6, 50), (11, 40), (3, 60), and (16, 30), you'll likely observe their relative positions on this graph. By looking at the chart, one can infer whether there is an upward, downward, or no discernible trend. An upward trend suggests the positive relationship; a downward trend indicates a negative relationship; scattered points with no clear direction suggest no correlation. For our case, through plotting and analyzing, one would see a pattern that might hint at a negative slope, indicating a potential negative correlation between \(x\) and \(y\).

• Scatter diagrams can reveal relationships such as linear, non-linear, or no apparent trend at all.
• Background grids on the diagram can help in assessing the slope and direction of the relationship.
• This visualization also aids in detecting outliers that might affect the relationship between the variables.

Sample covariance provides a quantifiable measure to assess the direction of a linear relationship between two variables. This statistic tells us how much two variables change together, signifying a pattern or mutual variance. To calculate sample covariance, we use the formula:
\[\text{Cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})\]

Here, \(\bar{x}\) and \(\bar{y}\) are the means of \(x\) and \(y\) respectively. For our data, calculating these gives us a covariance of \(-60\). This negative value indicates a negative relationship between the variables \(x\) and \(y\). In simpler terms, as \(x\) increases, \(y\) tends to decrease.

• A positive covariance implies that \(x\) and \(y\) increase together.
• A negative covariance signifies that when one variable goes up, the other tends to go down.
• A covariance close to zero suggests no linear relationship.

The magnitude of covariance isn't standardized, making it less intuitive to interpret compared to correlation coefficient. It gives us the direction but not the strength of the relationship.

The sample correlation coefficient, often denoted as \(r\), quantifies both the strength and direction of a linear relationship between two variables. It is a normalized version of covariance, ranging between -1 and 1. Calculating \(r\) involves the formula:
\[r = \frac{\text{Cov}(x, y)}{s_x s_y}\]

Where \(s_x\) and \(s_y\) are the standard deviations of \(x\) and \(y\). For our data, with previously calculated values, \(r \approx -0.973\). This close to -1 value strongly indicates a robust negative linear relationship, implying a pronounced inverse relation as \(x\) rises, \(y\) generally declines.

The correlation coefficient provides several insights:

• Values of \(r\) close to 1 or -1 suggest a strong relationship, be it positive or negative.
• A value of 0 implies no linear relationship.
• Sign (+ or -) indicates the direction of the relationship: + for a positive trend, - for a negative trend.
• Correlations tell about linear associations but may not imply causation.

This simple yet comprehensive statistic allows for objective comparisons across different data sets and is fundamental in predictive analysis and statistical modeling.