Abstract
Chapter 3 was a coherent presentation of material on descriptive statistics and probability distributions, which are viewed as theoretical populations. Chapter 4 follows with seven sections that are grouped into three themes. The first four sections treat inferential statistics, beginning with an introductory section that introduces the central question of whether a test population is the same as or different from a general population. This is followed by a section that presents the Cramer–von Mises test for consistency of a data set with a normal distribution. This test serves as a convenient analytical tool for observing the convergence as n → ∞ of distributions of means of samples of size n to a normal distribution, regardless of the underlying distribution from which the samples are drawn. Section 4.3 presents the central limit theorem, which is then used in Section 4.4 to address statistical inference. Section 4.5 serves as a brief introduction to maximum likelihood, and the chapter closes with two sections on conditional probability. Many of the problems use real data sets that were introduced in Chapterer 3.
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Notes
- 1.
Of course there has to be some difference between ABE members and other students, or else we wouldn’t be able to count them. Until proven otherwise, the difference is assumed to be some distinctive but meaningless marking.
- 2.
See Problem 3.2.6.
- 3.
See Section 3.4.
- 4.
See Example 4.3.3.
- 5.
Standard cutoff values are arbitrary. There is no particular reason why the cutoff used in most fields should be 5 % rather than 4 %, and yet most papers on statistical analysis make definitive judgments based on this arbitrary choice. In a sense, statistics is a way of codifying judgments based on probability; in many cases it is better scientific practice to base these judgments on context rather than fixed arbitrary rules.
- 6.
See Problem 4.1.2.
- 7.
Most authors use the formula J(x) ∕ n for all x and change the definition of J from “less than” to “less than or equal to.” This has the drawback of requiring an explanation for why the formula involving i is the one that is actually used in statistical calculations. The distinction between the definitions is otherwise meaningless.
- 8.
Pronounced “MY-zees.”
- 9.
There is a dizzying number of tests for goodness of fit to a distribution type. The most commonly used is probably the Anderson–Darling test, which is similar to the Cramer–von Mises test except that it weights discrepancies in the tails of the distributions higher than discrepancies near the mean. Most experts consider Anderson–Darling to be slightly better, but the difference seems too small to compensate for the extra complexity, given that statistical tests merely produce numerical values that must be interpreted in context. From a modeler’s point of view, there is an inherent advantage to using methods that are easy to understand. See [11] for a more complete treatment of normality testing.
- 10.
This is roughly the square root of one-tenth.
- 11.
See Section 4.4.
- 12.
See [12] for a treatment that uses the best methods and distinguishes a variety of subcases.
- 13.
Pronounced “show-veh-NAY.”
- 14.
Whereas the Cramer–von Mises test is at least arguably as good as any other test for distribution type, Chauvenet’s test is no longer used. Our purpose here is to explore the issue of outliers; in this context, simplicity is better than correctness. Most statisticians favor the Grubbs test, which appears in Problems 4.2.12–4.2.14.
- 15.
We want to determine how much oxygen a particular Olympic swimmer uses in 200- m training swims, but our equipment is only available at one pool. The “population” in this case consists of the different 200- m swims for the single athlete.
- 16.
This issue is addressed in Section 4.4.
- 17.
See Problem 4.3.1.
- 18.
See Section 3.7.
- 19.
The reporter did not identify the source of this claim. It certainly was not in the research paper that inspired the climate change news story.
- 20.
For example, does an individual have a heart attack in a given period or not?
- 21.
See Theorem 4.3.3.
- 22.
See Problem 4.4.2.
- 23.
It is a common fallacy to interpret a correlation in terms of causation; for example, we might think that a correlation between high dopamine levels and psychosis means that too much dopamine makes a person psychotic. This may be true, but demonstrating a correlation is not enough to support the claim. It is also possible that being psychotic raises dopamine levels, or that some other condition is responsible for both the psychosis and the high level of dopamine. A good rule of thumb is that causation should not be inferred from correlation unless only one of the possibilities of “A causes B” and “B causes A” is plausible. For example, if people who eat carrots have better night vision than people who don’t, we can safely conclude that something in the carrots is responsible for the improved vision, as it seems clearly implausible that being able to see well at night makes a person hungry for carrots.
- 24.
See Section 4.1 for a discussion of why this detail is necessary.
- 25.
See Theorem 4.3.2.
- 26.
See Theorem 3.6.1.
- 27.
See Problem 3.4.6 for background and commentary.
- 28.
See Theorem 3.4.1.
- 29.
Pearson drew the same conclusion using his more sophisticated chi-square test. He also suggested an explanation: large numbers were more likely than small numbers because the pips on the dice reduced the weight of each face; hence, the faces with larger numbers were lighter than those with smaller numbers. See [3] for a further discussion.
- 30.
Theorem 4.3.3.
- 31.
The real situation is far more complicated than this simplified narrative suggests and serves as a lesson on the dangers of using a result for a real case that is true only in some limit. The example that follows hints at the problem, which is explored in more detail in Problem 4.4.10.
- 32.
See Problems 4.3.8 and 4.4.10.
- 33.
Mark Twain said, “There are three kinds of lies: white lies, damn lies, and statistics.” This appears to be an example of Twain’s dictum. The apparent coincidence is not entirely random, because the person looking for a coincidence has deliberately chosen the consecutive string of presidents that makes the strongest statistical case. Add to that the general principle that left-handedness is merely one of many possible coincidences in the list of presidents, and it should not be surprising that some coincidence can be found.
- 34.
See Theorem 3.4.1.
- 35.
See Section 3.6.
- 36.
See [9], for example.
- 37.
See Section 3.2.
- 38.
- 39.
Because an event can have only one outcome, we can use the single term “event” to represent a single outcome as well as an event consisting of multiple outcomes.
- 40.
Recall that the event C c is the set of outcomes not in C.
- 41.
Recall that | S | is the number of elements in the set S.
- 42.
See Section 3.2.
- 43.
See Equation (3.2.3).
- 44.
Nearly all the subjects were white and middle class, so it is not clear that the sample of subjects was representative of the population as a whole.
- 45.
Full analysis of the test would have required all of the subjects to be tested using venous blood. The value of the study is somewhat limited by this not having been done. On the other hand, it would have been difficult to get parents to agree to the additional test.
- 46.
“dL” is a deciliter, equal to 100 cubic centimeters.
- 47.
Remember to use the probabilistic definitions of these terms rather than the meanings associated with normal language use. An experiment is any observation of an outcome, so we have an experiment consisting of the observation of the condition and an experiment consisting of the observation of the test result. The “condition experiment” is a hypothetical experiment that results in knowledge of the presence or absence of the condition.
- 48.
Minimally, any three of them will suffice. The four events are mutually exclusive and all-encompassing; hence, their sum is 1 and any one of them is easily calculated from the other three.
- 49.
See Problem 4.7.1.
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Ledder, G. (2013). Working with Probability. In: Mathematics for the Life Sciences. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7276-6_4
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