91Ó°ÊÓ

Productivity Petralia \({ }^{15}\) studied the correlation between investment as a percent of GNP and productivity growth of nine countries: Belgium (B), Canada (C), France (F), Germany (G), Italy (I), Japan (J), the Netherlands (N), the United Kingdom (UK), and the United States (US). Productivity is given as output per empoyee-hour in manufacturing. The data they collected for the years \(1960-1976\) is given in the following table. \begin{tabular}{|c|ccccc|} \hline Country & US & UK & B & I & C \\ \hline\(x\) & 14 & 17 & 20.5 & 21 & 21.5 \\ \hline\(y\) & 2.2 & 3.3 & 6.4 & 5.8 & 3.8 \\ \hline Country & \(\mathrm{F}\) & \(\mathrm{N}\) & \(\mathrm{G}\) & \(\mathrm{J}\) & \\\ \hline\(x\) & 22 & 24 & 25 & 32 & \\ \hline\(y\) & 5.7 & 6.3 & 5.8 & 9.0 & \\ \hline \end{tabular} Here \(x\) is investment as percent of GNP, and \(y\) is the productivity growth (\%). a. Determine the best-fitting line using least squares and the correlation coefficient. b. What does this model predict the productivity growth will be when investment is \(30 \%\) of GNP? c. What does this model predict investment as a percent of GNP will be if productivity growth is \(7 \% ?\)

Step by step solution

01

Calculate the Mean of x and y

First, we'll calculate the mean of the investment percentages, \( x \), and the productivity growth percentages, \( y \). This is done by adding up all the values for each variable and then dividing by the number of values (9 in this case).\[x_{mean} = \frac{14 + 17 + 20.5 + 21 + 21.5 + 22 + 24 + 25 + 32}{9}\]\[x_{mean} = 21.55\]\[y_{mean} = \frac{2.2 + 3.3 + 6.4 + 5.8 + 3.8 + 5.7 + 6.3 + 5.8 + 9.0}{9}\]\[y_{mean} = 5.3667\]

02

Calculate the Covariance and Variance

We calculate the covariance of \( x \) and \( y \) and the variance of \( x \).Covariance \( S_{xy} \):\[S_{xy} = \sum{(x_i - x_{mean})(y_i - y_{mean})}\]Variance \( S_{xx} \):\[S_{xx} = \sum{(x_i - x_{mean})^2}\]After calculation:\[S_{xy} = 131.3 \, \quad S_{xx} = 234.45\]

03

Determine the Slope and Intercept of the Line

Using the covariance and variance, we calculate the slope (\( b \)) and the y-intercept (\( a \)) of the best-fit line.\[b = \frac{S_{xy}}{S_{xx}} = \frac{131.3}{234.45} \approx 0.5602\]\[a = y_{mean} - b \times x_{mean} = 5.3667 - 0.5602 \times 21.55 \approx -6.7335\]

04

Write the Equation of the Line

The equation of the best-fitting line (line of regression) is given by:\[y = bx + a\]Substitute the values of \( a \) and \( b \):\[y = 0.5602x - 6.7335\]

05

Calculate the Correlation Coefficient

The correlation coefficient \( r \) is calculated using:\[r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}\]Where \( S_{yy} \) is the variance of \( y \). First, calculate \( S_{yy} \):\[S_{yy} = \sum{(y_i - y_{mean})^2} = 43.06\]Then compute \( r \):\[r = \frac{131.3}{\sqrt{234.45 \times 43.06}} \approx 0.941\]

06

Predict Productivity Growth for 30% Investment

Using the line equation, predict the productivity growth \( y \) for \( x = 30 \):\[y = 0.5602 \times 30 - 6.7335 = 10.0725y \approx 10.1\%\]

07

Predict Investment for 7% Productivity Growth

To find \( x \) when productivity growth \( y = 7 \)%:\[7 = 0.5602x - 6.7335\]Solve for \( x \):\[x = \frac{7 + 6.7335}{0.5602} \approx 24.3y \approx 24.3\%\]

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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, often denoted as \( r \), measures the strength and direction of a linear relationship between two variables. It ranges from \(-1\) to \(1\). A value close to \(1\) indicates a strong positive linear relationship, meaning as one variable increases, so does the other. Conversely, a value near \(-1\) suggests a strong negative relationship, where one variable increases as the other decreases. A value around zero indicates little to no linear relationship.

In the exercise, the correlation coefficient was computed to be approximately \(0.941\). This suggests a strong positive linear relationship between investment as a percentage of GNP and productivity growth. This high \( r \) value means we can confidently use the linear model to predict how changes in investment percentages might affect productivity growth.

Least Squares Method

The least squares method is a statistical technique used to determine the best-fitting line through a set of data points. The goal is to minimize the sum of the squares of the vertical distances (residuals) between the observed values and the values predicted by the line.

• First, we calculate the mean values for both variables, \( x \) and \( y \).
• Next, we find the covariance between \( x \) and \( y \) and the variance of \( x \).
• The slope \( b \) of the line is then calculated as \( \frac{S_{xy}}{S_{xx}} \).
• The y-intercept \( a \) is found using \( y_{mean} - b \times x_{mean} \).

In the provided exercise, these steps led to the equation \( y = 0.5602x - 6.7335 \). This equation can be used to predict one variable given the other, assuming the linear relationship holds.

Productivity Growth

Productivity growth is an important economic indicator that measures the increase in efficiency of production. Specifically, in the context of this exercise, it is defined as the change in output per employee-hour within the manufacturing sector. Growth in productivity is crucial for economic development as it often leads to higher wages, better standards of living, and increased competitiveness globally.

In terms of this data set, productivity growth is expressed as a percentage and acts as our dependent variable \( y \). The analysis and predictions made using the linear regression model aim to understand how investments related to GNP influence productivity changes over time.

Investment as a Percentage of GNP

Investment as a percentage of Gross National Product (GNP) serves as a measure of how much of a country's economic output is being reinvested into capital goods. These investments include infrastructure, machinery, and technology, which are crucial for sustaining and enhancing economic productivity in the long term.

In this exercise, the percentage of GNP investment is used as the independent variable \( x \). By examining how this variable correlates with productivity growth (\( y \)), economists can hypothesize about the potential impact of investment policies. The linear relationship derived in the analysis provides insights into how increasing or decreasing investments might affect productivity, thus helping in economic planning and policy making.