91Ó°ÊÓ

Productivity Recall from Example 3 that Cohen \(^{16}\) studied the correlation between corporate spending on communications and computers (as a percent of all spending on equipment) and annual productivity growth. In Example 3 we looked at his data on 11 companies for the period from 1985 to \(1989 .\) The data found in the following table is for the years \(1977-1984\) \begin{tabular}{|l|llllll|} \hline\(x\) & 0.03 & 0.07 & 0.10 & 0.13 & 0.14 & 0.17 \\ \hline\(y\) & -2.0 & -1.5 & 1.7 & -0.6 & 2.2 & 0.3 \\ \hline\(x\) & 0.24 & 0.29 & 0.39 & 0.62 & 0.83 & \\ \hline\(y\) & 1.3 & 4.2 & 3.4 & 4.0 & -0.5 & \\ \hline \end{tabular} Here \(x\) is the spending on communications and computers as a percent of all spending on equipment, and \(y\) is the annual productivity growth. Determine the best-fitting line using least squares and the correlation coefficient.

Step by step solution

01

Organize the Data

We have two lists for the variables: spending percentage (\(x\)) and productivity growth (\(y\)). They are given as follows: \[ x = [0.03, 0.07, 0.10, 0.13, 0.14, 0.17, 0.24, 0.29, 0.39, 0.62, 0.83] \] \[ y = [-2.0, -1.5, 1.7, -0.6, 2.2, 0.3, 1.3, 4.2, 3.4, 4.0, -0.5] \] Now, we need to compute the necessary statistics to determine the best-fitting line.

02

Calculate Means and Sums

Calculate the means of \(x\) and \(y\), the sum of \(x^2\), the sum of \(y^2\), and the sum of the product \(xy\). \[ \bar{x} = \frac{\sum x_i}{N}, \quad \bar{y} = \frac{\sum y_i}{N} \] where \(N = 11\). Compute \(\sum x, \sum y, \sum x^2, \sum y^2, \) and \(\sum xy \).

03

Compute the Slope (b) and Intercept (a)

Using the formulas for the slope \(b\) and intercept \(a\) of the least-squares line: \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{\sum xy - N \cdot \bar{x} \cdot \bar{y}}{\sum x^2 - N \cdot (\bar{x})^2} \] \[ a = \bar{y} - b \cdot \bar{x} \] Substitute the calculated sums and means to find \(b\) and \(a\).

04

Write the Equation of the Line

The least-squares regression line is given by \( y = a + bx \). Substitute the calculated values of \(a\) and \(b\) into the equation.

05

Calculate the Correlation Coefficient (r)

The correlation coefficient \(r\) is calculated using: \[ r = \frac{N \cdot \sum xy - \sum x \cdot \sum y}{\sqrt{(N \cdot \sum x^2 - (\sum x)^2)(N \cdot \sum y^2 - (\sum y)^2)}} \] Using the computed sums from step 2, calculate \(r\).

06

Interpret the Correlation Coefficient

The value of \(r\) indicates the strength and direction of the linear relationship between \(x\) and \(y\). An \(r\) close to 1 or -1 suggests a strong correlation, while an \(r\) near 0 suggests a weak correlation.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient, denoted as \( r \), is a statistical measure that calculates the strength and direction of a linear relationship between two variables. In our exercise, these variables are spending on communications and computers (\( x \)), and annual productivity growth (\( y \)). The formula to calculate \( r \) is:\[ r = \frac{N \cdot \sum xy - \sum x \cdot \sum y}{\sqrt{(N \cdot \sum x^2 - (\sum x)^2)(N \cdot \sum y^2 - (\sum y)^2)}}\]Here, \( N \) represents the total number of data points.

To interpret \( r \):

• An \( r \) value of 1 indicates a perfect positive linear relationship.
• An \( r \) value of -1 indicates a perfect negative linear relationship.
• An \( r \) value near 0 suggests a weak linear relationship.

It’s important to note that correlation does not imply causation. While \( r \) can show how strongly two variables are related, it does not indicate why or the driving forces behind their relationship.

Data Analysis

In data analysis, organizing raw data into structured forms is key. This process involves collecting, cleaning, and interpreting data to extract useful insights. For our exercise, we have a dataset containing two variables: spending percentage (\( x \)) and productivity growth (\( y \)).

The first step in data analysis is organizing and cleaning the data. This involves ensuring the data is clear of errors and correctly formatted. Next, we calculate statistics such as mean, sum of squares, and cross products. These statistics are essential for performing deeper analysis like finding trends and correlations.

• Mean: It represents the average value of a dataset, providing a central value.
• Sum of Squared Values: Useful to determine variance, showing how data is spread out.
• Cross Product: It helps in calculating the correlation between two datasets.

Through these calculations, we can identify any underlying patterns or relationships between variables in the dataset, aiding informed decision-making.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In our scenario, linear regression helps us derive the best-fitting line for the data on company spending and productivity growth.

This line is expressed by the equation \( y = a + bx \), where \( b \) is the slope and \( a \) is the y-intercept.

• Slope \( b \): Shows the rate at which \( y \) changes for a unit change in \( x \). A positive \( b \) indicates an upward trend, while a negative \( b \) signals a downward trend.
• Intercept \( a \): This is the point where the line crosses the y-axis, representing the value of \( y \) when \( x = 0 \).

By using the least squares method, we minimize the distance between the data points and the linear line. This helps in predicting future trends and making decisions based on past data. Linear regression not only quantifies relationships but also can be used in forecasting and identifying trends across different fields.