Which test would you choose?

Suppose that today you have a test about a book that you were supposed to read. The book has 100 chapters, but you only had time to read 75 of them. (You were too busy watching How I Met Your Mother, making lists, driving to Sonic, sleeping, or blogging to finish the rest of the chapters. We don't really have time to debate about whether or not these activities were the best use of your time.) You walk into the test very nervous (because you only know 75% of what you're supposed to know), and your teacher (drumroll) has two stacks of DIFFERENT tests. She says she is going to let each student choose which test they want to take. She tells you your options and you analyze them.

Option #1 is a test with 100 fill-in-the-blank questions, one question about each chapter. Since you read 75% of the chapters, are adequately intelligent, and the answers are obvious if you read the chapters they are about, this option will guarantee you a grade of 75 on the test. So your expected grade if you choose this option is:
EV(option #1) = .75 x 100 = 75

Option #2 is a test with only one question, about a random chapter. Since you read 75% of the chapters, are adequately intelligent, and the answers are obvious if you read the chapters they are about, with this option you have a 75% chance of getting a 100 on the test, but a 25% chance that you will get a 0. So your expected grade if you choose this option is:
EV(option #2) = .75 x 100 + .25 x 0 = 75

Either way, your expected grade is a 75, so you should be indifferent about which option you choose. 

Before continuing with the conclusion of this post, I would like to make a quick note about an objection to my model that an adequately intelligent reader may have. I was thinking that the model might be different for a multiple choice test because if you took the long test, then you would have a 25% chance of answering correctly each of the 25 questions that you didn't know, thus raising your expected grade for the long test to 81.25. But, the expected value for the short test would also increase because if the single question was about a chapter you didn't read, you would have a 25% chance of getting it right. In the case of a multiple-choice test, the expected values of both options would be 81.25 (still equal).

However, there are other factors to consider.

First, what is your level of risk-aversion? A very risk-averse individual would prefer a guaranteed 75. But a risk-lover would hope for 100 even if it means she might get a 0. I will refer to this as the risk-aversion effect.

Next, how much do you value the time you will waste taking the test? If each question takes one minute to complete, the short test will only take one minute while the long test will take 100 minutes to complete. I will refer to this as the time-wasting effect.

A stereotypical extreme b.a. (badas*, not necessarily bachelor of arts) is a risk-lover and  thinks tests are a huge waste of time (she would rather be riding her motorcycle), so would choose the short test.

A stereotypical extreme dork is risk-averse and enjoys taking tests (she learns for fun so would probably use spare time to read anyway), and would thus choose the long test.

For a more moderate student that is neither of the two extremes, both the risk-aversion effect and the time-wasting effect will be present, so one must look at her individual personality to see which effect overpowers the other.

I myself am a risk-averse individual (I wouldn't gamble high-stakes and uncertainty stresses me out), so the risk-aversion effect is present. However, I am also an economist, and the opportunity cost of those 99 extra minutes wasted taking the long test could have been put to much better use (such as riding a motorcycle or reading the last 25 chapters), so the time-wasting effect is also present.

So the real question isn't "Which test would you choose?" but rather "What is your individual personality?"

Are we children?

You may have noticed that I started my last post with "As a kid, between 2% and 90% of my thoughts were consumed with thinking of products..." which would lead one to infer that I am no longer a child. I apologize for being unclear with my wording, because the question of whether or not I am a child is still up in the air.

Not knowing where to begin, I looked up "child" in the dictionary.
child [chahyld] noun 1. a person between birth and full growth.

I would agree that this definition matches conventional views of childhood, but it is in the interpretation and application that we notice discrepancy. I know I'm a person. I'm also pretty sure that I am after birth, but have I reached full growth? What is "full growth?" It's when you're done growing.That means you're stuck being exactly the same as you are now for every day after today, like a machine that makes tubs of lard or something, repeating the same motion again and again, never changing. Full growth does not sound fun to me, because it means I will never learn anything new, never have realizations that change my life or experience any exciting life-changing events.

To illustrate this idea of "full growth," I have included a graph for your convenience. When you are at full growth, ripeness (could also be "maturity" if that helps you conceptualize it, but not being mature means you're immature and I don't want to portray a negative connotation) has reached a  maximum and thus the change in ripeness ("maturation") is zero. As you'll notice, in the green part of the graph, ripeness is increasing at an increasing rate while in the purple part of the graph ripeness is increasing at a decreasing rate. This means that in the green part of the graph, the change in ripeness increases each year while in the purple part of the graph the change in ripeness decreases each year.

 In the past four years, while I may not have changed much in height, I have changed more in other ways than in any other four-year period of my life. (...wow that sounds really sentimental) Four years ago I was a naive high school student that thought a 70 on a Pre-Calc final was the end of the world, was afraid to talk to people unless it was about Harry Potter (although I still love talking about Harry Potter), and  wondered why a wall would shake when two people were together in the room next door. Besides learning to live on my own, I am now a college junior that knows there are more important measures of a person's quality than grades, has the confidence to introduce herself to random individuals just because they look like they might be witty, and still has a lot to learn about the structure of buildings and why the walls would shake so unexpectedly. I anticipate that in the next four years of my life, I will change even more as I learn about the structure of buildings as well as build my life.

So I assert that I am not close to approaching "full growth" because I am still changing at an increasing rate and thus on the green part of the graph. People are dynamic individuals that never stop changing, so full growth is more like an asymptote, and I will never reach full growth. If a child is someone that is between birth and full growth, then a child I will always be.

How do you make quality oatmeal?

As a kid, between 2% and 90% of my thoughts were consumed with thinking of products that are commonly called by their brand name and not their actual name (such as Chapstick is really lip balm, Vaseline is really petroleum jelly, Kleenex is really tissue, Band-Aid is really adhesive bandage, Listerine is really mouth wash, to name a few...after 21 years of thinking I have quite an extensive list). A portion of the other 10% to 98% of my thoughts were observing subtle differences between my parents. For example, my mom tied shoes by looping one lace around the other while my dad tied shoes by making a knot with two bunny ears. My mom cut the core out of apples by making two nicks on the sides and then scooping out the middle while my dad just made two sharp slices in a V shape. My mom made oatmeal that was thick, chunky and lumpy so that the consistency was similar to vomit while my dad made oatmeal that was watery and severely undercooked.

Needless to say, neither of these styles of oatmeal were very appealing to me, so I quickly learned to make my own oatmeal just the way I like it. As you probably could have guessed based on the precedent I have set with my previous posts, the question I am actually going to address has nothing to do with the quality of oatmeal. What I would actually like to question is why I noticed differences and not similarities. I never noticed that both of my parents cut apples with their right hands, they both didn't put spoons in the microwave when they were cooking the oatmeal, and they both have noses.

This shows that we often take similarities for granted. For example, I bet you noticed that unlike other colleges, Geneseo is situated on a steep hill, but took for granted that all college students write papers. I bet you noticed that the kid behind you in Philosophy 100 has hair in a ponytail down to his butt, but you may not have noticed that the color bears a striking resemblance to your own shade. I bet you noticed the last time your roommate's actions were different from your wishes (they didn't do the dishes), but took for granted all of the times their actions were similar to what you wanted (they took out the trash).

Instead of being annoyed that we go to a school on a hill, we could unite with college students across the world about writing twenty page papers about obscure topics. Instead of thinking that long-haired kid is weird, we could get some advice on the best shampooing strategies for chestnut-brown-haired individuals. Instead of getting mad that our roommates neglect the dishes, we could be grateful that they take out the trash. Instead of being picky about oatmeal consistency, we would learn not to put metal spoons in the microwave. So, maybe if we stopped noticing all these differences, and started noticing similarities, the world would be a happier place (or at least have less fires in the microwaves).

What makes us so productive some days and so unproductive other days?

I got a lot of work done today. I even wrote in my blog! We all have some days in which we are quite productive and other days that we don't accomplish anything.

So the dependent variable, y, is a measure of productivity.

One possible independent variable that would explain 100% (R squared=1) of the variation in y would be my level of motivation. This proves that a high R squared doesn't necessarily mean the model is a useful one, because this model isn't very useful. Of course I'm more motivated when I get more work done. But why? What I'd really like to find out is: what causes me to be more motivated? By explaining the variation in my motivation level, I can use this knowledge to turn lazy days into productive ones.

So I came up with a list of other possible independent variables to test:
SLEEP= # of hours of sleep the night before
JOG= # of miles I jogged in the morning
ZUMBA=a dummy variable equal to 1 if I had Zumba and 0 otherwise
LIST= a dummy variable equal to 1 if I made a to-do list and 0 otherwise
FACEBOOK= # of minutes spent on facebook
WINE= a dummy variable equal to 1 if I drank the night before and 0 otherwise

There are many problems with these possible variables.
SLEEP: While it may be true that to an extent more sleep leads to higher productivity, this doesn't really apply with too much sleep (for example, 13 hours makes me feel like I woke up out of a black hole and I don't want to accomplish anything). This could be a problem we observe when trying to apply data taken on developing world countries to the United States. For example, eating cheeseburgers in some African countries makes them "healthier" because any calories are better than no calories. But eating cheeseburgers where we have an abundance of food makes us less healthy.

JOG: This variable will likely show a high correlation with my productivity level, but it is questionable whether my productivity is what causes me to jog, or that jogging causes my productivity. This is a reason we should question statistics such as "eating breakfast makes you healthier." But maybe people eat breakfast because they are health conscious, rather than being health conscious because you eat breakfast. When put this way, the statistic doesn't make as much sense.

FACEBOOK: The number of minutes spent on Facebook is just a representation of how many minutes I didn't spend in the library instead. What actually leads to productivity is doing work in the library, rather than not going on facebook. This is similar to the commonly accepted statistic that replacing milk with soda increases bone density. Is it the increase in milk or the decrease in soda that actually contributes to higher bone density?

LIST: Maybe before I make the list, I am destined to be productive because I have a lot to do, and making the to do list merely indicates that I have to be productive, rather than actually causes productivity. For this reason, we should question statistics such as "reading to your baby in the womb will make them smarter." Maybe you read to your baby because you are smart, so they are destined to be smart whether you read to them or not. Reading to your baby is an indicator of smart parents, and smart parents are the actual cause of smart babies, not reading to them in the womb.

ZUMBA: I could arguably try to fudge this statistic so that it appears that Zumba increases my productivity. For example, I could only teach Zumba on weekdays, and then on weekends when I am less productive I didn't have Zumba. I would leave out the variable WINE so that it appears that when I have Zumba I am more productive, when the real reason is that I didn't go out the night before. That way, people will think Zumba causes productivity and want to come to my class. I could have an ulterior motive behind this study. This can be seen when cereal companies tell you that eating breakfast helps you loose weight or dairy companies tell you that milk decreases your chance of osteoporosis.

Can you think of possible issues with WINE? What about possible issues with other statistics you have heard? What leads to higher productivity is a question that still needs to be answered. But I do hope that I have encouraged you to question statistics that you would normally accept without question. What may appear to be true may not be true at all.