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to estimate would be the proportion of cases in the population suffering from the disease. If we obtain the value of 0.01 in the sample, for instance, we use this to estimate the true population proportion. You might think, at first glance, that point estimation is a great thing. However, we alluded to earlier why it can be problematic. How this is so is best exemplified by an example. Suppose you would like to catch a fish and resort to very primitive ways of doing so. You obtain a long stick, sharpen the end of it into a spear, and attempt to throw the spear at a fish wallowing in shallow waters. The spear is relatively sharp, so if it “hits” the fish, you are going to secure the catch. However, even intuitively, without any formality, you know there is a problem with this approach. The problem is that catching the fish will be extremely difficult, because even with a good throw, the probability of the spear hitting the fish is likely to be extremely small. It might even be close to zero. This is even if you are a skilled fisherperson!

      So, what is the solution? Build a net of course! Instead of the spear, you choose instead to cast a net at the fish. Intuitively, you widen your probability of catching the fish. This idea of widening the net in this regard is referred to in statistics as interval estimation. Instead of estimating with merely a point (sharp spear), you widen the net in order to increase your chances of catching the fish. Though interval estimation is a fairly broad concept, in practice, one of the most common interval estimators is that of a confidence interval. Hence, when we compute a confidence interval, we are estimating the value of the parameter, but with a wider margin than with a point estimator. Theoretically at least, the margin of error for a point estimator is equal to zero because it allows for no “wiggle room” regarding the location of the parameter. So, what is a good margin of error? Just as the significance level of 0.05 is often used as a default significance level, 95% confidence intervals are often used. A 95% confidence interval has a 0.95 probability of capturing (or “covering”) the true parameter. That is, if you took a bunch of samples and on each computed a confidence interval, 95% of them would capture the parameter. If you computed a 99% interval instead, then 99% of them would capture the parameter.

      The following is a 95% confidence interval for the mean for a z-distribution,

      ȳ – 1.96σ_M < μ < ȳ + 1.96σ_M

where ȳ is the sample mean and σ_M is the standard error of the mean, and, when unpacked, is equal to (we will discuss this later). Notice that it is the population mean, σ/√n, that is at the center of the interval. However, μ is not the random variable here. Rather, the sample on which ȳ was computed is the random sample. The population parameter μ in this case is assumed to be fixed. What the above confidence interval is saying, probabilistically, is the following:

      Over all possible samples, the probability is 0.95 that the range between ȳ – 1.96σ_M and ȳ + 1.96σ_M will include the true mean, μ.

      Now, it may appear at first glance that increasing the confidence level will lead to a better estimate of the parameter. That is, it might seem that increasing the confidence interval from 95% to 99%, for instance, might provide a more precise estimate. A 99% interval looks as follows:

      ȳ – 2.58σ_M < μ < ȳ + 2.58σ_M

      Notice that the critical values for z are more extreme (i.e. they are larger in absolute value) for the 99% interval than for the 95% one. But, shouldn’t increasing the confidence from 95% to 99% help us “narrow” in on the parameter more sharply? At first, it seems like it should. However, this interpretation is misguided and is a prime example of how intuition can sometimes lead us astray. Increasing the level of confidence, all else equal, actually widens the interval, not narrows it. What if we wanted full confidence, 100%? The interval, in theory, would look as follows:

      ȳ – ∞σM < μ < ȳ + ∞σM

      That is, we are quite sure the true mean will fall between negative and positive infinity! A truly meaningless statement. The morale of the story is this – if you want more confidence, you are going to have to pay for it with a wider interval. Scientists usually like to use 95% intervals in most of their work, but there is definitely no mathematical principle that says this is the level one should use. For some problems, even a 90% interval might be appropriate. This again highlights the importance of understanding research principles, so that you can appreciate why the research paper features this or that level of confidence. Be critical (in a good way). Ask questions of the research paper.

      1.5 Essential Philosophical Principles for Applied Statistics

As already briefly discussed, one reason why many students (especially those outside of the mathematical sciences) have an initial disdain for learning statistics is that they have the impression that statistics is mathematics, and since they dislike mathematics, they naturally believe they will dislike statistics. However, it is important to realize that statistics is not mathematics. Even a deeply mathematical version of statistics (the so-called field of mathematical statistics) is still, by itself, not mathematics! It is statistics, first and foremost. The range of mathematics used in the communication of statistical concepts varies greatly from source to source, just as it does from course to course in college. An analogy can be drawn to physics. Physics itself is not mathematics. Mathematics, however, is the most useful vehicle for communicating ideas in physics. For instance, the idea of the instantaneous rate of change is not, by itself, mathematics. However, it is very well defined in mathematics as the derivative. Historically it began as a concept, and evolved eventually over thousands of years into a mathematical expression so we could actually work with the philosophical concept. If this challenges your view on what mathematics actually is, that is a good thing! Most people who do not know better associate mathematics with numbers and equations. Yes, it’s that too, but it’s so much more. It is not number-crunching, it is a discipline of concepts rigorized for the sake of communication and manipulation, and agrees a lot (but not always) with the physical world.

      Even more important than essential mathematics is probably the essential philosophical principles that underlie scientific and statistical analyses. You may ask what on earth a discussion of philosophy has to do with statistics and science? Everything! Now, I am not talking about the kind of philosophy where we question who we are and the meaning of life, lay in bed as Descartes did, and eventually conclude “I think, therefore I am,” wear a robe, smoke a pipe, and contemplate our own existences. Most empirically-trained scientists are practicing an empirical philosophy. Philosophy is at the heart of all scientific and non-scientific disciplines. “Philosophy of science” is a branch of philosophy that, loosely put, surveys the assumptions of science and what it means to establish evidence in the sciences. Translated, it means “thinking about what you are doing, rather than just doing it.” Philosophy of science asks questions such as the following:

       What does it mean to establish scientific evidence and why is scientific evidence so valued vs. other forms of evidence?

       Why is the experiment usually considered the gold-standard of scientific evidence, and why is correlational research often not nearly as prized?

       Why is science inductive rather than deductive? What are some of the problems associated with induction that make drawing scientific conclusions so difficult?

       Is establishing causation possible, and if so, what does it mean to establish it?

       Why has science adopted a statistical approach to establishing evidence in most of its fields? What is so special about the statistical approach? If statistics can be so misleading, would we be better off not using them at all? Does the use of statistics advance science or hinder it?

       Do multivariate statistics help clarify or otherwise confuse and impede the search for scientific evidence? Do procedures such as factor

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