need in the near future probably should not be invested too heavily in stocks because during brief (i.e., small) periods of time, the value of stocks can decline, sometimes precipitously. However, money you do not need for a longer period of time probably is better invested in stocks than in bonds.
Now that we have learned some reasons why we tend not to incorporate statistical information into our thinking, let’s distinguish among three concepts that we’ve already implicitly touched on and that are foundational for statistical thinking: a population, a variable, and a sample. These are not complicated distinctions, but they are critical in this course and in being better consumers of statistical information. A population is the entire group of people you want to draw conclusions about. In our example of stock market and bond market gains, the population would be every year since 1928. In our example of where to eat on Friday night, the population would be every person who’s eaten at that restaurant. All members of a population must have some characteristic in common. In research, such a characteristic is called a variable. A variable is a quality that has different values or changes in the population. For instance, qualities such as height, age, personality, happiness, and intelligence each differ among people; hence, each is a variable. Variables can also be environmental features, such as classroom wall color or investment returns.
Photo 1.8 Stocks generally rise in value. The highlighted pockets of time note the exceptions to this historical trend.
Source: ©Macrotrends LLC/“Down Jones – 100 Year Historical Chart”
For most research studies, and for most situations in life more generally, it is impossible to examine or be familiar with each and every member of the population. Therefore, we make particular observations from the population, and based on that sample, we draw conclusions about the population (see Figure 1.1). One year could be a sample of stock market and bond market gains (or depending on the one year in question, losses). Your roommate’s friend is the sample in that example. Had you read reviews of the restaurant, doing so would have provided a larger sample of data on which to base your decision on where to eat dinner.
Figure 1.1 Relationship Between and Purposes of a Population and a Sample
Population: entire group of people you want to draw conclusions about.
Variable: characteristic that has different values or changes among individuals.
Sample: subset of people from the population that is intended to represent the characteristics of the larger population.
Let’s return to two other previous examples. Regarding casinos advertising the people who won money gambling at their facilities, the population would be all people who gamble in casinos. The sample would be the winners that are in the advertisements. Based only on this sample, it appears that winning money in casinos is normal (which of course it is not; it is just the opposite, actually). In our example of the base-rate fallacy and the “surfer-looking” student at my college, the population would be all students at my college, 90% of whom are from either Michigan, Ohio, or Indiana. The sample is the one student who looks like he is a surfer, which is something we associate with people from California or Florida more than with people from these other three states. He was one of about 1,500 students in the population. Given his appearance, it would be easy to assume he was from somewhere other than what is indicative of the population he was a member of.3
Learning Check
1 Because algorithms guarantee a correct solution to a problem, why do people tend to prefer heuristics in their thinking?A: Heuristics provide us with answers to our problems more quickly than do algorithms. In addition, heuristics are not necessarily going to lead us astray (despite the tone of much of what you’re reading). So, we keep using them because they are efficient and sometimes correct.
2 We are far more likely to die in a car accident than in an airplane accident. Yet, people tend to fear flying more than driving. Explain why people fear flying more than driving even though driving is more dangerous.A: When a plane crashes, especially a commercial passenger plane, we hear about it on the news perhaps for several days after it happened. However, we are less likely to hear about car accidents on the news. Therefore, because of the availability heuristic, flying is often feared more than is driving even though driving is far more dangerous.
3 Vicks® VapoRubTM (Procter & Gamble, Cincinnati, OH) is medicine you can spread on your chest to help break up congestion. To me, it smells a lot like menthol cough drops. So as a kid when I had a cold, I ate the Vicks VapoRub. Use the representativeness heuristic to explain why I ate the VapoRub instead of spreading it across my chest as the product is supposed to be used.A: We make mental categories of things that seem to “go together.” Here, both cough drops and VapoRub have a menthol smell (at least to me), so given that one is supposed to eat cough drops, I figured anything with a similar smell is supposed to be eaten as well.
4 Explain why people are more likely to carry their umbrellas when they hear there is a “20% chance of rain” than when there is an “80% chance it will be dry.” Both phrases contain the same information, so why is there a difference in how people respond to them?A: This is an example of the framing effect. By hearing the word “rain,” people think about getting wet. By hearing the word “dry,” people do not think about getting wet. Therefore, the image of being wet prompts people to carry their umbrellas.
5 LeBron James, one of the best players in professional basketball, has challenged me to a game of one-on-one basketball. By using information about the law of small numbers, explain how I could maximize the likelihood that I beat James at his sport.A: Even the best basketball players will sometimes miss a shot. Even the worst basketball players will sometimes make a shot. Therefore, to maximize my chances of winning, which on the surface seem nonexistent, I would want to play just one shot against LeBron James. This is an application of the law of small numbers. Perhaps he’ll miss his shot and I will make mine and, thus, win. The longer the game goes, the more likely I am to lose because he is the better basketball player.
6 Suppose I made you the following offer: You pay me $4, and I will flip a legitimate coin for which you call “heads” or “tails.” If you call the flip correctly, I’ll pay you $10. If you call it incorrectly, you get nothing. How many times, if any, would you play this game with me? Explain your reasoning.A: If you want to make a lot of money, you should play this game as often as possible. On average, you will call the flip correctly 50% of the time. When you do, you win $10. When you don’t, you lose $4. Suppose you played this game twice, winning once and losing once. To play twice costs $8 ($4 each time). If you win only once, you walk away with $10, so you made $2. The more you play this game, the more money you will make. However, we know from the law of small numbers that you want to play it more than once or twice because it is possible you could end up losing money with a small number of flips. But over the course of many flips, you will win money.
7 What is the difference between a population and a sample?A: The population is larger than the sample. The sample is used to draw conclusions about the population, which is the entire group you want to learn about in a research study.
8 Why is blood pressure a variable?A: A variable is a characteristic that differs among members of a population. People have different blood pressure levels, so therefore it is a variable (and one of great interest to medical researchers).
Misunderstanding Connections Between Events
In addition to not using information about probability, we as human beings also have a need to perceive order in the world. Think about being at a party, and you suddenly find yourself around people you do not know. You know almost nothing about them (other than they are at the same party you are), so what do you say to them? You have almost no idea