Problem Set - Statistics & Models

1. Statistics block FiveThirtyEight is considered one of the best sites for election predictions. In 2016, it was the most correct site in predicting the victory of Donald Trump in the presidential election. This is FiveThiryEight's explanation for how they got it right:

Based on what most of us would have thought possible a year or two ago, the election of Donald Trump was one of the most shocking events in American political history. But it shouldn’t have been that much of a surprise based on the polls — at least if you were reading FiveThirtyEight. Given the historical accuracy of polling and where each candidate’s support was distributed, the polls showed a race that was both fairly close and highly uncertain...

.... So why did our model — using basically the same data as everyone else — show such a different result? ... We think the outcome — and particularly the fact that Trump won the Electoral College while losing the popular vote — validates important features of our approach. More importantly, it helps to explain why Trump won the presidency.

Clinton was leading in the vast majority of national polls, and in polls of enough states to get her to 270 electoral votes, although her position in New Hampshire was tenuous in the waning days of the campaign. So there wasn’t any reasonable way to construct a polling-based model that showed Trump ahead...

But people mistake having a large volume of polling data for eliminating uncertainty. It doesn’t work that way. Yes, having more polls helps to a degree, by reducing sampling error and by providing for a mix of reasonable methodologies. Therefore, it’s better to be ahead in two polls than ahead in one poll, and in 10 polls than in two polls. Before long, however, you start to encounter diminishing returns. Polls tend to replicate one another’s mistakes: If a particular type of demographic subgroup is hard to reach on the phone, for instance, the polls may try different workarounds but they’re all likely to have problems of some kind or another. The cacophony of headlines about how “CLINTON LEADS IN POLL” neglected the fact that these leads were often quite small and that if one poll missed, the others potentially would also...

... Historically, meanwhile, the error is larger in state polls than in national polls. That’s because there’s less of an opportunity for polling errors to cancel each other out. Suppose, for example, that the polls underestimate Clinton’s performance with Hispanic voters, but overestimate it among white voters without college degrees. In national polls, the overall effect might be relatively neutral. But the state polls will err in opposite directions, overestimating Clinton’s performance in states with lots of noncollege white voters but underestimating it in Hispanic-heavy states.

That’s something like what happened this year. In fact, the error in national polls wasn’t any worse than usual. Clinton was ahead by 3 to 4 percentage points in the final national polls... ...But what about the state polls? They were all over the place. Clinton actually overperformed FiveThirtyEight’s adjusted polling average in 11 states and the District of Columbia. The problem is that these states were California, Hawaii, Illinois, Massachusetts, Nevada, New Jersey, New York, New Mexico, Oregon, Rhode Island and Washington. Since all of these states except for Nevada and perhaps New Mexico were already solidly blue, that only helped Clinton to run up the popular vote margin in states whose electoral votes she was already assured of.

2. The Upshot column of the New York Times in September, 2017 was titled, The Economic Case for Letting Teenagers Sleep a Little Later, and included the following information:

A Brookings Institution policy brief investigated the trade-offs between costs and benefits of pushing back the start times of high school in 2011. It estimated that increased transportation costs would most likely be about $150 per student per year. But more sleep has been shown to lead to higher academic achievement. They found that the added academic benefit of later start times would be equivalent to about two additional months of schooling, which they calculated would add about $17,500 to a student’s earnings over the course of a lifetime. Thus, the benefits outweighed the costs.

And

A recent analysis by the RAND Corporation goes much further.

Marco Hafner, Martin Stepanek and Wendy Troxel conducted analyses to determine the economic implication of a universal shift of middle and high school start times to 8:30 a.m. at the earliest. This study was stronger than the Brookings one in a number of ways. It examined each state individually, because moving to 8:30 would be a bigger change for some than for others. It also looked at changes year by year to see how costs and benefits accrued over time. It examined downstream effects, like car accidents, which can affect lifetime productivity. And it considered multiplier effects, as changes to the lives of individual students might affect others over time.

They found that delaying school start times to 8:30 or later would contribute $83 billion to the economy within a decade. The gains were seen through decreased car crash mortality and increased student lifetime earnings.

3. The Upshot column of the New York Times in April 2016 examined the role of money on educational outcomes for public schools across the United States and produced it as this graphic:

4. The Upshot column of the New York Times in December 2017 examined the effectiveness of school districts comparing the exam scores of Third Graders to that of Eighth Graders. This is the graphic from the article: