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Chapter 4: Problem 20

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{llllll} \hline x & 0 & 5 & 7 & 8 & 9 \\ \hline y & 3 & 8 & 6 & 9 & 4 \end{array} $$

Short Answer

Expert verified
The correlation coefficient is 0.44, indicating a weak positive linear relationship between \(x\) and \(y\).

Step by step solution

01

- Draw the Scatter Diagram

Plot each pair \(x, y\) on a coordinate plane. The points to plot are: (0, 3), (5, 8), (7, 6), (8, 9), and (9, 4). This provides a visual representation of the relationship between the variables.
02

- Calculate Means of x and y

Find the mean of the x-values and y-values. \ \[ \bar{x} = \frac{0 + 5 + 7 + 8 + 9}{5} = 5.8 \ \bar{y} = \frac{3 + 8 + 6 + 9 + 4}{5} = 6 \ \]
03

- Calculate the Sum of Products \((x_i - \bar{x})(y_i - \bar{y})\)

Compute \(x_i - \bar{x}\) and \(y_i - \bar{y}\) for each pair, then multiply them and sum up. \ \ \( \sum (x_i - \bar{x})(y_i - \bar{y}) = (0 - 5.8)(3 - 6) + (5 - 5.8)(8 - 6) + (7 - 5.8)(6 - 6) + (8 - 5.8)(9 - 6) + (9 - 5.8)(4 - 6) = (-5.8)(-3) + (-0.8)(2) + (1.2)(0) + (2.2)(3) + (3.2)(-2) = 17.4 - 1.6 + 0 + 6.6 - 6.4 = 16 \ \ \)
04

- Calculate the Sum of Squares \((x_i - \bar{x})^2\)

Compute \(x_i - \bar{x}\) square for each x-value, then sum up. \ \ \( \sum (x_i - \bar{x})^2 = (0 - 5.8)^2 + (5 - 5.8)^2 + (7 - 5.8)^2 + (8 - 5.8)^2 + (9 - 5.8)^2 = 33.64 + 0.64 + 1.44 + 4.84 + 10.24 = 50.8 \)
05

- Calculate the Sum of Squares \((y_i - \bar{y})^2\)

Compute \(y_i - \bar{y}\) square for each y-value, then sum up. \ \ \( \sum (y_i - \bar{y})^2 = (3 - 6)^2 + (8 - 6)^2 + (6 - 6)^2 + (9 - 6)^2 + (4 - 6)^2 = 9 + 4 + 0 + 9 + 4 = 26 \)
06

- Compute the Correlation Coefficient

Use the formula for the correlation coefficient \(r\). \ \ \( r = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sqrt{ \sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2 } } = \frac{16}{ \sqrt{50.8 \cdot 26} } = \frac{16}{ \sqrt{1320.8} } = \frac{16}{36.32} = 0.44 \)
07

- Determine Linear Relationship

Interpret the correlation coefficient. Since \(r = 0.44\), which is not close to \(1\) or \(-1\), there is a weak positive linear relationship between \(x\) and \(y\).

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Key Concepts

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

scatter diagram
A scatter diagram is a powerful visual tool that helps us understand the relationship between two variables. Imagine a basic graph with one variable plotted along the x-axis and the other along the y-axis. By plotting individual data points (each corresponding to a pair of values), we can visually inspect any patterns or trends. For example, in this specific exercise, we plot pairs such as (0, 3) and (5, 8). When all pairs are plotted, the scatter diagram provides a clear picture of how the variables might be related.
linear relationship
A linear relationship is a type of association where changes in one variable are directly proportional to changes in another. We assess this by calculating the correlation coefficient. If this coefficient, denoted as 'r', is close to 1 or -1, it indicates a strong linear relationship. In contrast, values near 0 hint at a weaker or non-linear relationship. For the given data (0, 3), (5, 8), (7, 6), (8, 9), and (9, 4), we derived a correlation coefficient of 0.44, suggesting a weak positive linear relationship.
sum of squares
The sum of squares is a crucial step in determining the correlation coefficient. It measures the variance or spread of data points around the mean value. To calculate this, we first find the average (mean) of our x and y values. Then, for each value, we subtract the mean and square the result. Summing all these squared differences gives us the sum of squares. For x-values in this problem, the sum of squares is \(50.8\); for y-values, it's \(26\). This provides insights into the overall variability in the data.
mean calculation
To calculate the mean, add together all the values of a given variable and then divide by the number of values. For our exercise, the mean of the x-values \( \bar{x} \) is calculated as follows: \[ \bar{x} = \frac{0 + 5 + 7 + 8 + 9}{5} = 5.8 \]. Similarly, the mean of the y-values \( \bar{y} \) is: \[ \bar{y} = \frac{3 + 8 + 6 + 9 + 4}{5} = 6 \]. Finding these means is an essential intermediate step in calculating the sum of squares and the correlation coefficient.
sum of products
The sum of products involves multiplying the deviations of each pair of variables from their respective means and then summing the results. We first subtract the mean from both x-values and y-values (e.g., \(x_i - \bar{x}\) and \(y_i - \bar{y}\)). Next, we multiply these deviations for each pair, and then sum these products. For our data, this calculation results in \( \textstyle \text{危} (x_i - \bar{x})(y_i - \bar{y}) = 16\). This metric helps us determine the direction and strength of the linear relationship between the variables.

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Most popular questions from this chapter

Suppose that two variables, \(X\) and \(Y\), are negatively associated. Does this mean that above-average values of \(X\) will always be associated with below- average values of \(Y ?\) Explain.

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{lllrrr} x & 2 & 4 & 6 & 6 & 7 \\ \hline y & 4 & 8 & 10 & 13 & 20 \\ \hline \end{array}$$

What does it mean to say that the linear correlation coefficient between two variables equals \(1 ?\) What would the scatter diagram look like?

What does it mean to say two variables are positively associated? Negatively associated?

Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the student's age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: $$ \begin{array}{ll|ll|ll} \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } \\ \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} \\ \hline 18 & 4.2 & 19 & 5.1 & 22 & 2.1 \\ \hline 18 & 1.1 & 19 & 2.3 & 22 & 3.6 \\ \hline 18 & 4.6 & 20 & 1.7 & 24 & 5.4 \\ \hline 18 & 3.1 & 20 & 6.1 & 25 & 4.8 \\ \hline 18 & 5.3 & 20 & 3.2 & 25 & 3.9 \\ \hline 18 & 3.2 & 20 & 5.3 & 26 & 5.2 \\ \hline 19 & 2.8 & 21 & 2.5 & 26 & 4.2 \\ \hline 19 & 2.3 & 21 & 6.4 & 35 & 8.1 \\ \hline 19 & 3.2 & 21 & 4.2 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35,8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35,8.1) has on the regression line.
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