91Ó°ÊÓ

A scatter diagram is a type of graph used in statistics to display the relationship between two sets of data. In our exercise, the scatter diagram plots the five given observations where each point represents a pair of values for the two variables: \((x_{i}, y_{i})\). With \(x\) on the horizontal axis and \(y\) on the vertical axis, you can plot the points: (4, 50), (6, 50), (11, 40), (3, 60), and (16, 30).

This visual representation is essential to determine if there is a correlation between the variables. A scatter diagram helps:

• Identify the type of relationship, like positive, negative, or none.
• Observe how closely the data points cluster along a line or curve.
• Detect outliers that deviate significantly from the general trend.

After plotting, examine the diagram. If we spot a downward trend, it suggests a negative relationship, meaning as one variable increases, the other tends to decrease. In this exercise, the plotted points appear to suggest such a negative trend.

Covariance is a measure that indicates the extent to which two variables change together. In simpler terms, it tells us whether two variables tend to move in the same direction or in opposite directions.

To calculate the sample covariance in this scenario, you use the formula: \[S_{xy} = \frac{1}{n-1} \sum (x_i - \bar{x})(y_i - \bar{y})\]Here, \(n\) is the number of data points, \(x_i\) and \(y_i\) are individual data points, and \(\bar{x}\) and \(\bar{y}\) are the means of \(x\) and \(y\), respectively.

In our exercise:

• The mean of \(x\) is 8, and the mean of \(y\) is 46.
• The computed covariance \(S_{xy} = -40\).

The negative covariance value indicates that the variables \(x\) and \(y\) have a tendency to move inversely. When \(x\) increases, \(y\) tends to decrease and vice versa. However, it's important to note that covariance only shows the direction of the relationship, not its strength.

The correlation coefficient is a statistical measure that describes the strength and direction of a relationship between two variables. Unlike covariance, the correlation coefficient is standardized, ranging from -1 to 1, making it easier to interpret.

To find the correlation coefficient, we use the formula: \[r = \frac{S_{xy}}{S_x S_y}\]where \(S_{xy}\) is the covariance of \(x\) and \(y\), and \(S_x\) and \(S_y\) are the standard deviations of \(x\) and \(y\), respectively.

In this exercise:

• The standard deviation of \(x\) is \(\sqrt{22.5}\), and for \(y\), it is \(\sqrt{184}\).
• The correlation coefficient computed is approximately \(-0.62\).

This negative correlation coefficient value suggests a moderate negative relationship between \(x\) and \(y\). A value near -1 would indicate a strong negative linear relationship, while a value near 0 would indicate no linear relationship.