*Part I:*

Using technology (statistics software or your calculator) create a scatter plot of the data and describe the relationship between year and number of warheads.

*Part II:*

Identify any outliers (visually) and remove them from your data set. Using technology, run a regression on the remaining data points. Discuss the validity of your regression to predict the number of warheads per year. What are your concerns? Will your model make accurate predictions? Is your model "useful" at all.

The points 2002, 2009 & 2012 appear to be outliers in the x-direction and they have been removed as instructed. Regression analysis was completed, but the data is clearly non-linear (this is confirmed by the residual graphs as well). The model cannot be used to make accurate predictions because the linearity assumption is violated.

*Part III:*

Re-express the data (leave the outliers out!) to find the best linear model possible to explain the relationship and make predictions.

- Write the equation of the model and find the number of warheads in 2002
**as predicted by the model.**

- Discuss the validity of your model to predict the number of warheads per year; be sure to address r
^{2}.

The re-expressed data is much more linear; but it may not be "straight enough" based on significant curvature / non-randomness of the residual plots. That said, I would be much more comfortable using the re-expressed model to predict the number of warheads, especially between 1948 and 1960 where the graph appears most linear. It might be best to consider two separate regressions based on the original scatterplot's curvature...but that's beyond the scope of this class! The r-squared value is over 90%, so if we can say the data is str8-enough, more than 90% of the variability in the number of wareheads can be explained by the progression of years. - Interpret the slope and the intercept of your linear model in context.

**Slope**= 0.361735. For each year's increase, we expect that the Logarithm of the number of warheads will increase by 0.361735 (minimally-acceptable answer).

Our model predicts that each year the U.S. nuclear arsenal increased by between two and three warheads (better answer which requires solving the log!).

**Intercept**= -700.637. In year ZERO, the US nuclear stockpile had negative 700 warheads; in context this number is meaningless; it merely gives us a way to get into the scale of the data we're examining. - What are your concerns about your model? Do you believe your model will make accurate predictions? Is your model "useful" at all?

By examining the scatter plot, the data appear MORE linear than before, but the post-regression residuals still show significant curvature (non-randomness); cannot be certain that the linearity assumption has been met. That said, our model should do a decent job of estimating # of warheads during the Cold War period.

*Part IV:*

Clearly the U.S. never had as many warheads as your model predicted in Part III. In fact, the original data set contained the actual number of U.S. warheads in 2002. Explain, using your understanding of U.S. History, what the data set probably looks like through the years of missing data.

The data that's missing is from 1966 to 2000 or so; this set of dates corresponds roughly to the Cold War period of U.S. history where the two world super-powers (the USA and USSR) were engaged in an arms race and "M.A.D." (mutually-assured destruction) policies. Both nations were building their nuclear arsenal in an attempt to gain an advantage should war ever erupt. In the end, the economy of the USSR was unable to sustain the nearly exponential growth of their military and their economy crumbled, bringing an end to the Cold War under President Reagan. In all likelyhood, the data that's missing (probably due to security concerns during the post-Cold War era) continues in a downward, curved manner so the entire data set looks like a parabola. The data points eliminated early in this analysis are probably not outliers! The decreases likely are a result of the post-Cold War era arms reduction treaties between the US and USSR/Russia like SALT.

*Additional Historical Background/Perspective/Extension Discussions:*

Some argue that the massive expenditures during the Cold War were an absurd waste of national resources; others that these expenditures were necessary to prevent nuclear holocaust and protect democracy - priceless outcomes so the cost does not matter. The truth probably lies somewhere between these two arguments; but there can be no doubt that Cold War expenditures into the nuclear program helped spur development in technologies and industries that would not otherwise exist today.

- Weapons targeting required precise positioning of mobile (submarine-based) ballistic missiles, so the military developed and launched satellite positioning; today's GPS.

For more information about navigation, visit the Virtual Tour exhibit Navigation.

- The need for stealth, self-sufficiency, and the ability to patrol independently for long periods led to the perfection of safe nuclear plants that were the predecessors of atomic energy generation plants in use in the U.S. and around the world.

For more information about nuclear propulsion, visit the Virtual Tour exhibit Nuclear Propulsion.

- Accurate, reliable and secure communications were crucial to ensure that the President could control launch orders for deployed nuclear weapons. This requirement led to improvements in acoustics, equipment and satellite technology leading to the advent of the cell phone and satellite radio and television. Furthermore, control of complicated vessels, long-range nuclear weapons, nuclear plants and satellites necessitated significant advancements in computer technology.

For more information about command, control and communications, visit the Virtual Tour exhibit Electronics.

- Fear of the enemy's ability to destroy communications on a large scale by attacking a single, important computer station lead to development of the internet - accessible from various locations with the proper equipment.

For more information about the creation of the internet, visit the Smithsonian Yahoo site Birth of the Internet.

**22 TRILLION DOLLARS ($22,000,000,000,000)**. To help understand how much money that is, Click here for a slideshow!