91Ó°ÊÓ

Productivity Bernstein \({ }^{13}\) studied the correlation between productivity growth and gross national product (GNP) growth of six countries: France (F), Germany (G), Italy (I), Japan (J), the United Kingdom (UK), and the United States (US). Productivity is given as output per employeehour in manufacturing. The data they collected for the years \(1950-1977\) is given in the following table. \begin{tabular}{|c|cccccc|} \hline Country & US & UK & F & I & G & J \\ \hline\(x\) & 2.5 & 2.7 & 5.2 & 5.6 & 5.7 & 9.0 \\ \hline\(y\) & 3.5 & 2.3 & 4.9 & 4.9 & 5.7 & 8.5 \\ \hline \end{tabular} Here \(x\) is the productivity growth \((\%),\) and \(y\) is the GNP growth (\%). a. Determine the best-fitting line using least squares and the correlation coefficient. b. What does this model predict the GNP growth will be when the productivity growth is \(7 \% ?\) c. What does this model predict the productivity growth will be if the GNP growth is \(7 \% ?\)

Step by step solution

01

Understand the Problem

We are given data on the productivity growth and GNP growth for six countries. We need to establish a linear relationship between these two variables using the least squares method, determine the correlation coefficient, and use the model to make predictions.

02

Calculate Mean of Data

Calculate the mean of the productivity growth values \( x \) and the mean of the GNP growth values \( y \). These means will be used in subsequent calculations for the least squares line.\[ \bar{x} = \frac{2.5 + 2.7 + 5.2 + 5.6 + 5.7 + 9.0}{6} = 5.117 \]\[ \bar{y} = \frac{3.5 + 2.3 + 4.9 + 4.9 + 5.7 + 8.5}{6} = 4.966 \]

03

Calculate the Slope (b)

The formula to find the slope \( b \) of the line is:\[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]Calculate this sum:\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (2.5 - 5.117)(3.5 - 4.966) + (2.7 - 5.117)(2.3 - 4.966) + \ldots + (9.0 - 5.117)(8.5 - 4.966) = 36.417 \]Then calculate:\[ \sum (x_i - \bar{x})^2 = (2.5 - 5.117)^2 + (2.7 - 5.117)^2 + \ldots + (9.0 - 5.117)^2 = 37.017 \]Thus, the slope \( b \) is:\[ b = \frac{36.417}{37.017} = 0.983 \]

04

Calculate y-intercept (a)

The intercept \( a \) of the least squares line can be calculated using:\[ a = \bar{y} - b\bar{x} \]Substitute the known values:\[ a = 4.966 - 0.983 \cdot 5.117 = -0.069 \]

05

Write the Least Squares Line Equation

Using the slope and intercept, the equation of the best-fitting line is:\[ y = 0.983x - 0.069 \]

06

Calculate the Correlation Coefficient

The formula for the correlation coefficient \( r \) is:\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \]Calculate:\[ \sum (y_i - \bar{y})^2 = (3.5 - 4.966)^2 + (2.3 - 4.966)^2 + \ldots + (8.5 - 4.966)^2 = 38.067 \]Thus, the correlation coefficient is:\[ r = \frac{36.417}{\sqrt{37.017 \cdot 38.067}} = 0.986 \]

07

Predict GNP Growth for Productivity Growth of 7%

Use the least squares line equation to predict \( y \) when \( x = 7 \% \):\[ y = 0.983(7) - 0.069 = 6.822 \]Therefore, the predicted GNP growth is approximately \( 6.82 \% \).

08

Predict Productivity Growth for GNP Growth of 7%

Rearrange the least squares equation to solve for \( x \):\[ 7 = 0.983x - 0.069 \]\[ x = \frac{7 + 0.069}{0.983} = 7.212 \]Therefore, the predicted productivity growth is approximately \( 7.21 \% \).

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Key Concepts

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

Understanding the Least Squares Method

The least squares method is a way to find the best-fitting line through a set of data points. This line, known as the least squares line, minimizes the sum of the squared differences between the observed values and those predicted by the line. The primary aim is to determine a straight line that best represents the relations

• The formula for the slope (\(b\)) of the line is \[b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}.\]
• Next, we calculate the intercept (\(a\)) with the formula \[a = \bar{y} - b\bar{x}.\]
• The equation of the best-fitting line is given by \(y = bx + a\).

This approach minimizes the on average error since the sum of the vertical distances squared (from each point to the line) is minimized.
Understanding least squares is fundamental because it's the basis for statistical methods to find trends and make predictions from data.

Exploring Linear Regression

Linear regression is a statistical technique used to explore the relationship between two variables. It models this relationship using a linear equation, which can be visually represented as a line.

• In the context of this kernel exercise, we're interested in how productivity growth affects GNP growth.
• By plotting productivity growth % on the x-axis and GNP growth % on the y-axis, we use the linear regression line to determine the impact of productivity growth.
• The slope of the line (\(b\)) indicates the rate at which GNP growth changes with productivity growth; for every 1% increase in productivity growth, GNP changes accordingly.

Linear regression not only helps to identify the strength of the relationship (strong or weak) but also shows the direction (positive or negative) of the relation between the variables. The ultimate output is an equation that one can utilize to predict outcomes for given input values.

Predictive Modeling with a Linear Equation

Predictive modeling uses statistical techniques like linear regression to forecast future outcomes based on past data. Once we have the line of best fit, it acts as a predictive model to estimate new values.

• For the given data, the model gives an equation \(y = 0.983x - 0.069\) that can be used to project the GNP growth given any productivity growth percentage value.
• In step 7 of our solution, when productivity growth is forecasted to be 7%, we substitute \(x = 7\) in the equation to predict GNP growth (%), calculated as roughly 6.82%.
• Similarly, we can rearrange the equation to find the productivity growth if we're given the target GNP growth, aiding in strategic business and economic planning scenarios.

This predictive capability is valuable for making informed decisions and forecasts in areas ranging from supply chain management to economic policies, by interpreting and utilizing past trends in data.