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Chapter 3: Problem 46

Five observations taken for two variables follow. $$\begin{array}{c|ccccc} x_{i} & 6 & 11 & 15 & 21 & 27 \\ \hline y_{i} & 6 & 9 & 6 & 17 & 12 \end{array}$$ a. Develop a scatter diagram for these data. b. What does the scatter diagram indicate about a relationship between \(x\) and \(y ?\) c. Compute and interpret the sample covariance. d. Compute and interpret the sample correlation coefficient.

Short Answer

Expert verified
The scatter diagram suggests a positive correlation, which is confirmed by the covariance (26) and correlation (0.88).

Step by step solution

01

Plotting the Scatter Diagram

To develop a scatter diagram, plot each pair of observations \(x_i, y_i\) on a coordinate grid. The x-axis represents the values of \(x\) and the y-axis represents the values of \(y\). For this data, plot the points: (6,6), (11,9), (15,6), (21,17), and (27,12). Connect the points to visually assess any correlation.
02

Observing the Scatter Diagram

Observe the plotted points to assess any apparent relationship between \(x\) and \(y\). Look for general trends, such as points clustering around a line (indicating a potential linear relationship) or randomness (indicating no obvious correlation).
03

Calculating the Mean Values

Calculate the mean of \(x\) values: \bar{x} = \frac{6+11+15+21+27}{5} = 16\. Similarly, calculate the mean of \(y\) values: \bar{y} = \frac{6+9+6+17+12}{5} = 10\. These means will be used in further calculations.
04

Computing Sample Covariance

Use the formula for sample covariance: \text{Cov}(x,y) = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})\. Substituting the values:\[\text{Cov}(x,y) = \frac{1}{4}((6-16)(6-10) + (11-16)(9-10) + (15-16)(6-10) + (21-16)(17-10) + (27-16)(12-10))\].Calculating each term and summing them gives the covariance.
05

Decision on the Sample Covariance

Compute: \[ \text{Cov}(x,y) = \frac{1}{4}((-10)(-4) + (-5)(-1) + (-1)(-4) + (5)(7) + (11)(2)) = \frac{1}{4}(40 + 5 + 4 + 35 + 22) = 26 \].The positive covariance suggests a positive linear relationship between \(x\) and \(y\).
06

Computing the Sample Correlation Coefficient

Use the formula for correlation coefficient: \rho = \frac{\text{Cov}(x,y)}{s_x s_y}\, where \ s_x\ and \ s_y\ are the sample standard deviations of \ x \ and \ y \, respectively. First, calculate \ s_x\ and \ s_y\ using the standard deviation formula and then solve for \[\rho = \frac{26}{\sqrt{80} \cdot \sqrt{33.5}}\].
07

Decision on the Correlation Coefficient

Calculate \ s_x = \sqrt{\frac{(6-16)^2 + (11-16)^2 + (15-16)^2 + (21-16)^2 + (27-16)^2}{4}} = \sqrt{80}\, and \ s_y = \sqrt{\frac{(6-10)^2 + (9-10)^2 + (6-10)^2 + (17-10)^2 + (12-10)^2}{4}} = \sqrt{33.5}\. Substitute in the correlation formula: \[ \rho = \frac{26}{\sqrt{80} \cdot \sqrt{33.5}} \approx 0.882 \] Since the correlation coefficient is close to 1, it indicates a strong positive linear relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot
A scatter plot is a vital tool in statistical analysis. It visually represents the relationship between two variables by plotting data points on a two-dimensional graph. In our situation, we have two variables, x and y, with five data points each: (6,6), (11,9), (15,6), (21,17), (27,12). The x-axis represents the variable 'x', and the y-axis, 'y'.
To create a scatter plot, follow these steps:
  • Draw the axes: Identify and label the x and y-axes based on your data.
  • Plot each pair of (x, y) coordinates as unique points on the graph.
  • Look for trends: Examine the scatter plot for patterns or correlations among the points.
A well-analyzed scatter plot reveals different trends:
  • Linear correlation: Most points fall along a straight line.
  • No correlation: Points are randomly dispersed.
In our example, the scatter plot suggests a pattern that may demonstrate a positive correlation, as observed points trend upward as x increases.
Sample Covariance
Sample covariance measures how two variables change together. It tells us the direction of the linear relationship between them.
Covariance is calculated using the formula:\[ \text{Cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) \]where:
  • \(n\) is the number of data pairs.
  • \(x_i\) and \(y_i\) are individual data points for x and y.
  • \(\bar{x}\) and \(\bar{y}\) are the mean values for x and y.
In our problem, calculating each term and summing them gives a covariance of 26. A positive covariance value like this indicates a positive linear relationship between x and y. This means that as x increases, y tends to increase as well. It's vital to note, though, that covariance doesn't quantify the strength of this relationship and is dependent on the units of the variables.
Correlation Coefficient
The correlation coefficient, denoted as \( \rho \), quantifies the strength and direction of a linear relationship between two variables. Unlike covariance, it has a fixed range from -1 to 1.
The formula for the correlation coefficient is:\[ \rho = \frac{\text{Cov}(x, y)}{s_x s_y} \]where:
  • \( \text{Cov}(x, y)\) is the covariance between x and y.
  • \(s_x\) and \(s_y\) are the standard deviations of x and y.
Calculations for our dataset reveal a correlation coefficient of approximately 0.882. Because this value is close to 1, it suggests a strong positive linear relationship between variables x and y.
Key points:
  • A correlation close to 1 indicates a strong positive relationship.
  • A correlation close to -1 signals a strong negative relationship.
  • A correlation near 0 suggests no linear relationship.
Hence, the correlation coefficient gives a clearer picture of the linear relationship's strength, unaffected by the units of measurement.

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