{"id":226,"date":"2021-08-02T19:02:34","date_gmt":"2021-08-02T23:02:34","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/statspsych\/?post_type=chapter&p=226"},"modified":"2023-12-30T16:07:14","modified_gmt":"2023-12-30T21:07:14","slug":"chapter-4","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/statspsych\/chapter\/chapter-4\/","title":{"raw":"4. Probability, Inferential Statistics, and Hypothesis Testing","rendered":"4. Probability, Inferential Statistics, and Hypothesis Testing"},"content":{"raw":"

<\/a>4a. Probability and Inferential Statistics<\/h1>\r\n
video lesson<\/a><\/h6>\r\nIn this chapter, we will focus on connecting concepts of [pb_glossary id=\"233\"]probability<\/strong> [\/pb_glossary] with the logic of inferential statistics.\r\n\"The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.\"\r\n\u2014 Bertrand Russel (1872-1970)<\/span>\r\n\r\nThese notable quotes represent why probability<\/strong> is critical for a basic understanding of scientific reasoning.\r\n\r\n\"Medicine is a science of uncertainty and an art of probability.\"\r\n\u2014 William Osler (1849\u20131919)<\/span>In many ways, the process of postsecondary education is all about instilling a sense of doubt and wonder, and the ability to estimate probabilities<\/strong>. As a matter of fact, that essentially sums up the entire reason why you are in this course. So let us tackle probability<\/strong>.\r\n\r\nWe will be keeping our coverage of probability<\/strong> to a very simple level, because the introductory statistics we will cover rely on only simple probability<\/strong>. That said, I encourage you to read further on compound and conditional probabilities<\/strong>, because they will certainly make you smarter at real-life decision making. We will briefly touch on examples of how bad people can be at using probability<\/strong> in real life, and we will then address what probability has to do with inferential statistics. Finally, I will introduce you to the [pb_glossary id=\"235\"]central limit theorem[\/pb_glossary]<\/strong>. This is probably one of the heftiest math concepts in the course, but worry not. Its implications are easy to learn, and the concepts behind it can be demonstrated empirically in the interactive exercises.\r\n\r\nFirst, we need to define probability<\/strong>. In a situation where several different outcomes are possible, the probability of any specific outcome is a fraction or proportion of all possible outcomes. Another way of saying that is this. If you wish to answer the question, \u201cWhat are the chances that outcome would have happened?\u201d, you can calculate the probability<\/strong> as the ratio of possible successful outcomes to all possible outcomes.\r\n

<\/a>Concept Practice: define probability<\/a><\/h4>\r\nPeople often use the rolling of dice as examples of simple probability<\/strong> problems.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"371\"]\"Dice\" \"Dice\" by matsuyuki is licensed under CC BY-SA 2.0[\/caption]\r\n\r\nIf you were to roll one typical die, which has a number on each side from 1 to 6, then the simple probability of rolling a 1 would be 1\/6. There are six possible outcomes, but only 1 of them is the successful outcome, that of rolling a 1.\r\n

<\/a>Concept Practice: calculate probability<\/a><\/h4>\r\nAnother common example used to introduce simple probability<\/strong> is cards. In a standard deck of casino cards, there are 52 cards. There are 4 aces in such a deck of cards (Aces are the \u201c1\u201d card, and there is 1 in each suit \u2013 hearts, spades, diamonds and clubs.)\r\n\r\n\"\"\r\n\r\nIf you were to ask the question \u201cwhat is the probability<\/strong> that a card drawn at random from a deck of cards will be an ace?\u201d, and you know all outcomes are equally likely, the probability<\/strong> would be the ratio of the number of times one could draw and ace divided by the number of all possible outcomes. In this example, then, the probability<\/strong> would be 4\/52. This ratio can be converted into a decimal: 4 divided by 52 is 0.077, or 7.7%. (Remember, to turn a decimal to a percent, you need to move the decimal place twice to the right.)\r\n

<\/a>Concept Practice: calculate probability<\/a><\/h4>\r\nProbability<\/strong> seems pretty straightforward, right? But people often misunderstand probability<\/strong> in real life. Take the idea of the lucky streak, for example. Let\u2019s say someone is rolling dice and they get 4 6\u2019s in a row. Lots of people might say that\u2019s a lucky streak and they might go as far as to say they should continue, because their luck is so good at the moment! According to the rules of probability<\/strong>, though, the next die roll has a 1\/6 chance of being a 6, just like all the others. True, the probability<\/strong> of a 4-in-a-row streak occurring is fairly slim: 1\/6 x 1\/6 x 1\/6 x 1\/6. But the fact is that this rare event does not predict future events (unless it is an unfair die!). Each time you roll a die, the probability<\/strong> of that event remains the same. That is what the human brain seems to have a really hard time accepting.\r\n

<\/a>Concept Practice: lucky streak<\/a><\/h4>\r\nWhen someone makes a prediction attached to a certain probability (e.g. there is only a 1% chance of an earthquake in the next week), and then that event occurs in spite of that low probability<\/strong> estimate (e.g. there is actually an earthquake the day after the prediction was made)\u2026 was that person wrong? No, not really, because they allowed for the possibility. Had they said there was a 0% chance, they would have been wrong.\r\n\r\nProbabilities<\/strong> are often used to express likelihood of outcomes under conditions of uncertainty. Like Bertrand Russell said, wise people rarely speak in terms of certainties. Because people so often misunderstand probability<\/strong>, or find irrational actions so hard to resist despite some understanding of probability<\/strong>, decision making in the realm of sciences needs to be designed to combat our natural human tendencies. What we are discussing now in terms of how to think about and calculate probabilities<\/strong> will form a core component of our decision-making framework as we move forward in the course.\r\n\r\nNow, let\u2019s take a look at how probability<\/strong> is used in statistics.\r\n

<\/a>Concept Practice: area under normal curve as probability<\/a><\/h4>\r\nWe saw that percentiles are expressions of area under a normal curve. Areas under the curve can be expressed as probability<\/strong>, too. For example, if we say the 50th percentile for IQ is 100, that can be expressed as: \"If I chose a person at random, there is a 50% chance that they will have an IQ score below 100.\u201d\r\n\r\n \r\n\r\n\"\"\r\n\r\nIf we find the 84th percentile for IQ is 115 there is another way to say that \u201cIf I chose a person at random, there is an 84% chance that they will have an IQ score below 115.\u201d\r\n\r\n\"\"\r\n

<\/a>Concept Practice: find percentiles<\/a><\/h4>\r\nAny time you are dealing with area under the normal curve, I encourage you to express that percentage in terms of probabilities<\/strong>. That will help you think clearly about what that area under the curve means once we get into the exercise of making decisions based on that information.\r\n

<\/a>Concept Practice: interpreting percentile as probability<\/a><\/h4>\r\nProbabilities<\/strong>, of course, range from 0 to 1 as proportions or fractions, and from 0% to 100% when expressed in percentage terms. In inferential statistics, we often express in terms of probability<\/strong> the likelihood that we would observe a particular score under a given normal curve model.\r\n

<\/a>Concept Practice: applying probability<\/a><\/h4>\r\nAlthough I encourage you to think of probabilities<\/strong> as percentages, the convention in statistics is to report to the probability of a score as a proportion, or decimal. The symbol used for \u201cprobability of score\u201d is p<\/em>. In statistics, the interpretation of \u201cp<\/em>\u201d is a delicate subject. Generations of researchers have been lazy in our understanding of what \u201cp<\/em>\u201d: tells us, and we have tended to over-interpret this statistic. As we begin to work with \u201cp<\/em>\u201d, I will ask you to memorize a mantra that will help you report its meaning accurately. For now, just keep in mind that most psychologists and psychology students still<\/em> make mistakes in how they express and understand the meaning of \u201cp<\/em>\u201d values. This will take time and effort to fix, but I am confident that your generation will learn to do better at a precise and careful understanding of what statistics like \u201cp<\/em>\u201d tell us\u2026 and what they do not.\r\n\r\nTo give you a sense of what a statement of p<\/em> < .05 might mean, let us think back to our rat weights example.\r\n\r\n\"\"\r\n\r\nIf I were to take a rat from our high-grain food group and place it on the distribution of untreated rat weights, and if it placed at Z<\/em> = .9, we could look at the area under the curve from that point and above. That would tell us how likely it would be to observe such a heavy rat in the general population of nontreated rats -- those that eat a normal diet.\r\n\r\nThink of it this way. When we select a rat from our treatment group (those that ate the grain-heavy diet), and it is heavier than the average for a nontreated rat, there are two possible explanations for that observation. One is that the diet made him that way. As a scientist whose hypothesis is that a grain-heavy diet will make the rats weigh more, I\u2019m actually motivated to interpret the observation that way. I want to believe this event is meaningful, because it is consistent with my hypothesis! But the other possibility is that, by random chance, we picked a rat that was heavy to begin with. There are plenty of rats in the distribution of nontreated rats that were at least that heavy. So there is always some probability<\/strong> that we just randomly selected a heavier rat. In this case, if my treated rat\u2019s weight was less than one standard deviation above the mean, we saw in the chapter on normal curves that the probability<\/strong> of observing a rat weight that high or higher in the nontreated population was about 18%. That is not so unusual. It would not be terribly surprising if that outcome were simply the result of random chance rather than a result of the diet the rat had been eating.\r\n\r\nIf, on the other hand, the rat we measured was 2.5 standard deviations above the mean, the tail probability<\/strong> beyond that Z-score<\/strong> would be vanishingly small.\r\n\r\n\"\"\r\n\r\nThe probability<\/strong> of observing such a rat weight in the nontreated population is very low, so it is far less likely that observation can be accounted for just by random chance alone. As we accumulate more evidence, the probability<\/strong> they could have come at random from the nontreated [pb_glossary id=\"248\"]population[\/pb_glossary]<\/strong> will weigh into our decision making about whether the grain-heavy diet indeed causes rats to become heavier. This is the way probabilities<\/strong> are used in the process of [pb_glossary id=\"259\"]hypothesis testing[\/pb_glossary]<\/strong>, the logic of inferential statistics that we will look at soon.\r\n

<\/a>Concept Practice: statistics as probability<\/a><\/h4>\r\nNow that you have seen the relevance of probability<\/strong> to the decision making process that comprises inferential statistics, we have one more major learning objective: to become familiar with the central limit theorem<\/strong>.\r\n\r\nHowever, before we get to the central limit theorem<\/strong>, we need to be clear on the distinction between two concepts:\u00a0 [pb_glossary id=\"249\"]sample[\/pb_glossary]<\/strong> and [pb_glossary id=\"248\"]population[\/pb_glossary]<\/strong>. In the world of statistics, the population<\/strong> is defined as all possible individuals or scores about which we would ideally draw conclusions. When we refer to the characteristics, or parameters, that describe a population<\/strong>, we will use Greek letters.\u00a0A sample<\/strong> is defined as the individuals or scores about which we are actually drawing conclusions. When we refer to the characteristics, or statistics, that describe a sample<\/strong>, we will use English letters.\r\n\r\nIt is important to understand the difference between a population<\/strong> and a sample<\/strong>, and how they relate to one another, in order to comprehend the central limit theorem<\/strong> and its usefulness for statistics. From a population<\/strong> we can draw multiple samples<\/strong>. The larger sample<\/strong>, the more closely our sample<\/strong> will represent the population<\/strong>.\r\n\r\nThink of a Venn diagram.\u00a0There is a circle that is a <\/span>population<\/strong>. Inside that large circle, you can draw an infinite number of smaller circles, each of which represents a <\/span>sample<\/strong>.<\/span>\r\n\r\n\"\"\r\n\r\nThe larger that inner circle, the more of the population<\/strong> it contains, and thus the more representative it is.\r\n\r\n\"\"\r\n\r\nLet us take a concrete example. A population<\/strong> might be the depression screening scores for all current postsecondary students in Canada. A sample from that population<\/strong> might be depression screening scores for 500 randomly selected postsecondary students from several institutions across Canada. That seems a more reasonable proportion of the two million students in the population<\/strong> than a sample<\/strong> that contains only 5 students. The 500 student sample<\/strong> has a better shot at adequately representing the entire population<\/strong> than does the 5 student sample<\/strong>, right? You can see that intuitively\u2026 and once you learn the central limit theorem<\/strong>, you will see the mathematical demonstration of the importance of sample<\/strong> size for representing the population<\/strong>.\r\n\r\nTo conduct the inferential statistics we are using in this course, we will be using the normal curve model to estimate probabilities<\/strong> associated with particular scores. To do that, we need to assume that data are normally distributed. However, in real life, our data are almost never actually a perfect match for the normal curve.\r\n\r\nSo how can we reasonably make the normality assumption? Here\u2019s the thing. The central limit theorem<\/strong> is a mathematical principle that assures us that the normality assumption is a reasonable one as long as we have a decent sample<\/strong> size.\r\n\r\n\"\"\r\n\r\nAccording to the theorem, as long as we take a decent-sized sample<\/strong>, if we took many samples<\/strong> (10,000) of large enough size (30+) and took the mean each time, the distribution of those means will approach a normal distribution, even if the scores from each <\/span>sample<\/strong> are not normally distributed. To see this for yourself, take a look at the histograms shown on the right. The top histogram came from taking from a <\/span>population<\/strong> 10,000 <\/span>samples <\/strong>of just one score each, and plotting them on a histogram. See how it has a flat, or rectangular shape? No way we could call that a shape approximating a normal<\/span>\u00a0curve. Next is a histogram that came from taking the means of 10,000 samples<\/strong>, if each sample<\/strong> included 4 scores. Looks slightly better, but still not very convincing. With a sample<\/strong> size of 7, it looks a bit better. Once our sample<\/strong> size is 10, we at least have something pretty close. Mathematically speaking, as long as the sample<\/strong> size is no smaller than 30, then the assumption of normality holds. The other way we can reasonably make the normality assumption is if we know the population itself follows a normal curve. In that case, even if individual samples<\/strong> do not have a nice shaped histogram, that is okay, because the normality assumption is one apply to the population<\/strong> in question, not to the sample<\/strong> itself. <\/span>\r\n\r\nNow, you can play around with an online demonstration<\/a> so you can really convince yourself that the central limit theorem<\/strong> works in practice. The goal here is to see what sample<\/strong> size is sufficient to generate a histogram that closely approximates a normal curve. And to trust that even if real-life data look wonky, the normal curve may still be a reasonable model for data analysis for purposes of inference.<\/span>\r\n

<\/a>Concept Practice: Central Limit Theorem<\/a><\/h4>\r\n

<\/a>4b. Hypothesis Testing<\/h1>\r\n
video lesson<\/a><\/h6>\r\nWe are finally ready for your first introduction to a formal decision making procedure often used in statistics, known as [pb_glossary id=\"259\"]hypothesis testing[\/pb_glossary]<\/strong>.\r\n\r\nIn this course, we started off with descriptive statistics, so that you would become familiar with ways to summarize the important characteristics of datasets. Then we explored the concepts standardizing scores, and relating those to probability as area under the normal curve model. With all those tools, we are now ready to make something!\r\n\r\n[caption id=\"attachment_268\" align=\"aligncenter\" width=\"1024\"]\"\" \"(not my) toolbox\" by erix! is licensed under CC BY 2.0 \"Dovetail Dresser\" by Didriks is licensed under CC BY 2.0<\/em>[\/caption]\r\n\r\nOkay, not furniture, exactly, but decisions.\r\n\r\nWe are now into the portion of the course that deals with inferential statistics. Just to get you thinking in terms of making decisions on the basis of data, let us take a slightly silly example. Suppose I have discovered a pill that cures hangovers!\r\n\r\n\"Tiberius\"\r\n\r\nWell, it greatly lessened symptoms of hangover in 10 of the 15 people I tested it on. I am charging 50 dollars per pill. Will you buy it the next time you go out for a night of drinking? Or recommend it to a friend? \u2026 If you said yes, I wonder if you are thinking very critically? Should we think about the cost-benefit ratio here on the basis of what information you have? If you said no, I bet some of the doubts I bring up popped to your mind as well. If 10 out of 15 people saw lessened symptoms, that\u2019s 2\/3 of people \u2013 so some people saw no benefits. Also, what does \u201cgreatly lessened symptoms of hangover\u201d mean? Which symptoms? How much is greatly? Was the reduction by two or more standard deviations from the mean? Or was it less than one standard deviation improvement? Given the cost of 50 dollars per pill, I have to say I would be skeptical about buying it without seeing some statistics!\r\n\r\nOn this list is a preview of the basic concepts to which you will be introduced as we go through the rest of this chapter.\r\n
\r\n

Hypothesis Testing Basic Concepts<\/p>\r\n\r\n<\/header>\r\n

\r\n