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The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let \(x\) be the number of different research programs, and let \(y\) be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: $$ \begin{array}{l|rrrrrr} \hline x & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline y & 1.8 & 1.7 & 1.5 & 1.4 & 1.0 & 0.7 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420\), \(\Sigma y^{2}=11.83, \Sigma x y=113.8\), and \(r \approx-0.973 .\) (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for \(y=\) mean number of patents per program?

Step by step solution

01

Understand the Least-Squares Equation

The least-squares equation is used to model the relationship between two variables by minimizing the sum of the squares of the residuals. The equation is given by \(y = a + bx\), where \(a\) is the y-intercept and \(b\) is the slope of the line.

02

Calculate the Slope (b)

The formula for the slope \(b\) in the least-squares method is:\[b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]Given values are \(n = 6\), \(\Sigma x = 90\), \(\Sigma y = 8.1\), \(\Sigma xy = 113.8\), and \(\Sigma x^2 = 1420\). Substitute these values to find \(b\):\[b = \frac{6(113.8) - 90(8.1)}{6(1420) - 90^2}\]Calculate the values accordingly.

03

Calculate the Y-intercept (a)

Once \(b\) is known, calculate \(a\) using the formula:\[a = \frac{\Sigma y - b(\Sigma x)}{n}\]Use the computed \(b\) in Step 2 along with \(\Sigma y = 8.1\), \(\Sigma x = 90\), and \(n = 6\) to find \(a\).

04

Form the Least-Squares Equation

Substitute the computed values of \(a\) and \(b\) into the least-squares equation \(y = a + bx\). This step finalizes the model for prediction.

05

Use the Equation to Forecast for x=15

With the model \(y = a + bx\) developed, substitute \(x = 15\) into this equation to predict \(y\), the mean number of patents per program for a company with 15 research programs.

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Key Concepts

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

Least-Squares Equation

The least-squares equation is a fundamental tool in statistics used to determine the line that best fits a set of data. This method reduces the sum of the squared differences between observed and predicted values, ensuring the most precise estimates. The equation itself is structured as follows:

• \( y = a + bx \)

In this formula, \( a \) represents the y-intercept, showing where the line crosses the y-axis. The value \( b \) is the slope, which indicates how much \( y \) changes with a unit increase in \( x \). For the pharmaceutical data, this equation helps predict the number of patents per program based on the number of different research programs.

Research Productivity

In a business context, such as pharmaceutical research, productivity is key for innovation and financial success. Research productivity here is defined as the mean number of patents per research program. Companies need to balance investing in new programs while maintaining high productivity. When a company manages too many research programs, productivity may decrease, a phenomenon evident in the given data with fewer patents per program as the number of programs increases. Identifying the optimal number of programs to maximize productivity thus becomes crucial for strategic planning.

Correlation Coefficient

The correlation coefficient \( r \) is a measure that describes the direction and strength of the relationship between two variables. It ranges from -1 to 1.

• A value close to 1 indicates a strong positive relationship, where as one variable increases, so does the other.
• A value close to -1 indicates a strong negative relationship, implying that as one variable increases, the other decreases.
• A value near 0 indicates no linear relationship.

In the context of the available data, the correlation coefficient \( r \approx -0.973 \) suggests a very strong negative correlation. This means that as the number of research programs increases, the mean number of patents per program sharply decreases, confirming the potential issue of over-extension in research capabilities.

Pharmaceutical Research Analysis

Conducting a pharmaceutical research analysis using statistical methods like the least-squares equation provides insights into how resources are allocated and the implications on productivity. By understanding the relationship between research programs and outputs, companies can make informed decisions. It can help identify when a company is stretching its resources too thinly, hindering innovation. An analytical approach enables firms to optimize their research and development efforts, ensuring cost efficiency and a high output of valuable patents, which are essential for competitive advantage and advancement in the pharmaceutical industry.