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Chapter 4: Problem 6

For the data set $$ \begin{array}{llllll} \hline \boldsymbol{x} & 2 & 4 & 8 & 8 & 9 \\ \hline \boldsymbol{y} & 1.4 & 1.8 & 2.1 & 2.3 & 2.6 \\ \hline \end{array} $$ (a) Draw a scatter diagram. Comment on the type of relation that appears to exist between \(x\) and \(y\) (b) Given that \(\bar{x}=6.2, s_{x}=3.03315, \bar{y}=2.04, s_{y}=0.461519\) and \(r=0.957241\), determine the least-squares regression line. (c) Graph the least-squares regression line on the scatter diagram drawn in part (a).

Short Answer

Expert verified
The scatter plot shows a positive linear relationship. The least-squares regression line is \( y = 1.1365 + 0.14556x \).

Step by step solution

01

Create a Scatter Diagram

Plot the given data points \[ (2, 1.4), (4, 1.8), (8, 2.1), (8, 2.3), (9, 2.6) \] on a graph with the x-values on the horizontal axis and the y-values on the vertical axis. For each pair \((x, y)\), put a dot at the corresponding \((x,y)\) location.
02

Describe the Relationship

Observe the scatter diagram. The data points show an upward trend, indicating a positive linear relationship between \(x\) and \(y\).
03

Use Given Statistics for Calculation

Given:\[ \bar{x} = 6.2, \, s_{x} = 3.03315, \, \bar{y} = 2.04, \, s_{y} = 0.461519, \, r = 0.957241 \]Use the formula for the least-squares regression line: \[ y = a + bx \] where \[ b = r \left( \frac{s_{y}}{s_{x}} \right) \] and \[ a = \bar{y} - b\bar{x} \]
04

Calculate the Slope (\(b\))

Calculate the slope (\(b\)): \[ b = 0.957241 \left( \frac{0.461519}{3.03315} \right) \approx 0.14556 \]
05

Calculate the Intercept (\(a\))

Calculate the intercept (\(a\)): \[ a = 2.04 - (0.14556)(6.2) \approx 1.1365 \]
06

Form the Regression Line

Combine the slope and intercept to form the least-squares regression line: \[ y = 1.1365 + 0.14556x \]
07

Graph the Regression Line

Plot the regression line \(y = 1.1365 + 0.14556x\) on the scatter diagram created in Step 1.

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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 type of data visualization that displays the relationship between two numerical variables. In the context of our exercise, scatter plots help to visualize the relationship between the variables x and y by plotting the data points \[ (2, 1.4), (4, 1.8), (8, 2.1), (8, 2.3), (9, 2.6) \] on a graph. Each axis represents one variable: the horizontal axis generally represents the independent variable (x), and the vertical axis represents the dependent variable (y).
Creating a scatter plot involves placing a dot at the intersection of x and y values for each pair of data.
Once plotted, you can begin to see patterns that indicate relationships, like an upward trend meaning a positive relationship or a downward trend for a negative one.
Scatter plots are instrumental for understanding data trends at a glance.
correlation coefficient
The correlation coefficient, denoted as \( r \), quantifies the strength and direction of the linear relationship between two variables.
It ranges between -1 and 1:
  • 1 indicates a perfect positive linear relationship
  • -1 indicates a perfect negative linear relationship
  • 0 indicates no linear relationship
In our given data, the correlation coefficient \( r \) is 0.957241, indicating a very strong positive linear relationship between x and y.
This means, as x increases, y also tends to increase significantly.
The correlation coefficient is crucial for understanding how closely the data points fit a straight line.
least-squares regression line
The least-squares regression line is a method used in linear regression that minimizes the sum of the squared differences between observed and predicted values. The formula for the regression line is given by \[ y = a + bx \] where
  • \( b \) is the slope
  • \( a \) is the y-intercept

This line predicts the value of y for any given x. It's called the 'least-squares' line because it minimizes the distances (squared) between the observed points and the line.
In our example, the regression equation is found to be \[ y = 1.1365 + 0.14556x \]
By plotting this line onto our scatter plot, we can make predictions about y given x values based on the trend revealed by the data.
slope and intercept calculation
Calculating the slope (\( b \)) and intercept (\( a \)) is fundamental in determining the least-squares regression line.
The slope shows the rate of change in y with respect to a unit change in x, determined by the formula: \[ b = r \frac{s_{y}}{s_{x}} \] where
  • \( r \) is the correlation coefficient
  • \( s_{y} \) and \( s_{x} \) are the standard deviations of y and x, respectively

In our case: \[ b \text {is calculated as: } b = 0.957241 \frac{0.461519}{3.03315} \thickapprox 0.14556 \].
The intercept represents the value of y when x is 0 and is calculated using the formula: \[ a = \bar{y} - b \bar{x} \] where
  • \( \bar{y} \) is the mean of y
  • \( \bar{x} \) is the mean of x

Here, it鈥檚 calculated as: \[a = 2.04 - (0.14556 * 6.2) \thickapprox 1.1365 \].
Combining these gives us the equation for the regression line: \[ y = 1.1365 + 0.14556x \].

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Most popular questions from this chapter

Write a paragraph that explains the concept of correlation. Include a discussion of the role that \(x_{i}-\bar{x}\) and \(y_{i}-\bar{y}\) play in the computation.

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