The following table shows the annual revenues (in millions of dollars) of a pharmaceutical company over the period 1990 to 2011.

Year	1990	1991	1992	1993	1994	1995	1996	1997	1998	1999	2000
Revenue	830	791	842	891	953	1,036	1,050	1,070	1,127	1,272	1,386
Year	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011
Revenue	1,304	1,379	1,417	1,441	1,525	1,613	1,870	1,593	1,595	2,132	2,330

Variable	Linear Trend	Quadratic Trend	Cubic Trend
Intercept	642.7	797.312	648.7662
t	60.496	21.8665	91.726
t2t-squared	N/A	1.6796	−5.7477
t3t-cube	N/A	N/A	0.2153
MSE	18333.16	15079.84	13689
se	135.4	122.8	117
R2R-squared	0.8984	0.9205	0.9317
Adjusted R2R-squared	0.8933	0.9122	0.9203

Which of the following is a revenue forecast for 2012 found by the polynomial trend equation with the best fit??
Multiple Choice

About 2 billion and 34 million dollars

About 2 billion and 337 million dollars

About 2 billion and 189 million dollars

About 2 billion and 149 million dollars

To determine the revenue forecast for 2012 using the polynomial trend equation with the best fit, we need to consider the trend with the highest R-squared value. In this case, the cubic trend equation has the highest R-squared value (0.9317), indicating the best fit.

The cubic trend equation is given by:
\[ \text{Revenue} = 648.7662 + 91.726t - 5.7477t^2 + 0.2153t^3 \]

Here, \( t \) represents the number of years since 1990. For the year 2012, \( t = 2012 - 1990 = 22 \).

Let's substitute \( t = 22 \) into the cubic trend equation to find the forecasted revenue for 2012.

\[
\text{Revenue} = 648.7662 + 91.726(22) - 5.7477(22^2) + 0.2153(22^3)
\]

First, calculate each term separately:
1. \( 91.726 \times 22 = 2017.972 \)
2. \( 5.7477 \times 22^2 = 5.7477 \times 484 = 2783.9668 \)
3. \( 0.2153 \times 22^3 = 0.2153 \times 10648 = 2292.0944 \)

Now, substitute these values back into the equation:
\[
\text{Revenue} = 648.7662 + 2017.972 - 2783.9668 + 2292.0944
\]

Perform the arithmetic:
1. \( 648.7662 + 2017.972 = 2666.7382 \)
2. \( 2666.7382 - 2783.9668 = -117.2286 \)
3. \( -117.2286 + 2292.0944 = 2174.8658 \)

Therefore, the forecasted revenue for 2012 is approximately 2174.8658 million dollars, which can be rounded to about 2 billion and 175 million dollars.

Among the given choices, the closest match is:
- **About 2 billion and 189 million dollars**

Thus, the correct answer is:
- **About 2 billion and 189 million dollars**

Do you want details or have any questions?

Here are some related questions to consider:
1. How would the forecast change if the quadratic trend equation were used instead?
2. What is the significance of the adjusted R-squared value in the context of model fitting?
3. How does the Mean Squared Error (MSE) influence the choice of the best fit model?
4. Can you explain the difference between the R-squared and adjusted R-squared values?
5. What are the limitations of using polynomial trend equations for forecasting?
6. How do outliers affect the accuracy of trend equations?
7. What other forecasting methods could be used to predict future revenues?
8. How might changes in the pharmaceutical industry impact future revenue forecasts?

**Tip:** When analyzing data trends, always consider the context and possible external factors that could influence future outcomes.

Discover how to forecast annual revenues of a pharmaceutical company from 1990 to 2011 using polynomial trend analysis. Explore the cubic trend equation with detailed steps and calculations. Learn about regression analysis and its application in forecasting. Suitable for advanced learners interested in statistical modeling and trend analysis.

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