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Introduction to Basic Statistical Methodology

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CHAPTER 1 ~ Introduction ~

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Examples: Toasting time, Temperature settings, etc.… 3 What is “random variation” in the distribution of a population? POPULATION 1: Little to no variation (e.g., product manufacturing) In engineering situations such as this, we try to maintain “quality control”… i.e., “tight tolerance levels,” high precision, low variability. But what about a population of, say, people?

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POPULATION 1: Little to no variation (e.g., product manufacturing) Density What is “random variation” in the distribution of a population? POPULATION 2: Little to no variation (e.g., clones) 4 Most individual values ≈ population mean value Examples: Gender, Race, Age, Height, Drug Response (e.g., cholesterol level),… NOT REALISTIC!!! Very little variation about the mean!

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POPULATION 2: Little to no variation (e.g., clones)POPULATION 3: Much variation (more realistic) Examples: Gender, Race, Age, Height, Drug Response (e.g., cholesterol level),… What is “random variation” in the distribution of a population? Density 5 Much more variation about the mean!

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GLOBAL OPERATION DYNAMICS, INC. What are “statistics,” and how can they be applied to real issues? Example: Suppose a certain company insists that it complies with “gender equality” regulations among its employee population, i.e., approx. 50% male and 50% female. 6 To test this claim, let us select a random sample of n = 100 employees, and count X = the number of males. (If the claim is true, then we expect X 50.) etc. Sample size n partially depends on the power of the test, i.e., the desired probability of correctly rejecting a false null hypothesis ( 80%). The larger the n, the higher the power.

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GLOBAL OPERATION DYNAMICS, INC. What are “statistics,” and how can they be applied to real issues? Example: Suppose a certain company insists that it complies with “gender equality” regulations among its employee population, i.e., approx. 50% male and 50% female. 7 To test this claim, let us select a random sample of n = 100 employees, and count X = the number of males. (If the claim is true, then we expect X 50.) etc. X = 64 males (+ 36 females) Questions: If the claim is true, how likely is this experimental result? (“p-value”) Could the difference (14 males) be due to random chance variation, or is it statistically significant?

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The experiment in this problem can be modeled by a random sequence of n = 100 independent coin tosses (Heads = Male, Tails = Female). It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses: probability of obtaining at least 0 Heads away from 50 is = 1.0000 “certainty” probability of obtaining at least 1 Head away from 50 is = 0.9204 probability of obtaining at least 2 Heads away from 50 is = 0.7644 probability of obtaining at least 3 Heads away from 50 is = 0.6173 probability of obtaining at least 4 Heads away from 50 is = 0.4841 probability of obtaining at least 5 Heads away from 50 is = 0.3682 probability of obtaining at least 6 Heads away from 50 is = 0.2713 probability of obtaining at least 7 Heads away from 50 is = 0.1933 probability of obtaining at least 8 Heads away from 50 is = 0.1332 probability of obtaining at least 9 Heads away from 50 is = 0.0886 probability of obtaining at least 10 Heads away from 50 is = 0.0569 probability of obtaining at least 11 Heads away from 50 is = 0.0352 probability of obtaining at least 12 Heads away from 50 is = 0.0210 probability of obtaining at least 13 Heads away from 50 is = 0.0120 probability of obtaining at least 14 Heads away from 50 is = 0.0066 etc. 0 8 The =.05 cutoff is called the significance level. 0.0066 is called the p-value of the sample. Because our p-value (.0066) is less than the significance level (.05), our data suggest that the coin is indeed biased, in favor of Heads. Likewise, our evidence suggests that employee gender in this company is biased, in favor of Males....... …..from 0 to 100 Heads…..

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GLOBAL OPERATION DYNAMICS, INC. What are “statistics,” and how can they be applied to real issues? Example: Suppose a certain company insists that it complies with “gender equality” regulations among its employee population, i.e., approx. 50% male and 50% female. 9 To test this claim, let us select a random sample of n = 100 employees, and count X = the number of males. (If the claim is true, then we expect X 50.) etc. X = 64 males (+ 36 females) Questions: If the claim is true, how likely is this experimental result? (“p-value”) Could the difference (14 males) be due to random chance variation, or is it statistically significant? HYPOTHESIS EXPERIMENT OBSERVATIONS

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The experiment in this problem can be modeled by a random sequence of n = 100 independent coin tosses (Heads = Male, Tails = Female). It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses: probability of obtaining at least 0 Heads away from 50 is = 1.0000 “certainty” probability of obtaining at least 1 Head away from 50 is = 0.9204 probability of obtaining at least 2 Heads away from 50 is = 0.7644 probability of obtaining at least 3 Heads away from 50 is = 0.6173 probability of obtaining at least 4 Heads away from 50 is = 0.4841 probability of obtaining at least 5 Heads away from 50 is = 0.3682 probability of obtaining at least 6 Heads away from 50 is = 0.2713 probability of obtaining at least 7 Heads away from 50 is = 0.1933 probability of obtaining at least 8 Heads away from 50 is = 0.1332 probability of obtaining at least 9 Heads away from 50 is = 0.0886 probability of obtaining at least 10 Heads away from 50 is = 0.0569 probability of obtaining at least 11 Heads away from 50 is = 0.0352 probability of obtaining at least 12 Heads away from 50 is = 0.0210 probability of obtaining at least 13 Heads away from 50 is = 0.0120 probability of obtaining at least 14 Heads away from 50 is = 0.0066 etc. 0 10 The =.05 cutoff is called the significance level. 0.0066 is called the p-value of the sample. Because our p-value (.0066) is less than the significance level (.05), our data suggest that the coin is indeed biased, in favor of Heads. Likewise, our evidence suggests that employee gender in this company is biased, in favor of Males....... ANALYSIS CONCLUSION

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“Classical Scientific Method” Hypothesis – Define the study population... Hypothesis – Define the study population... What’s the question? What’s the question? Experiment – Designed to test hypothesis Experiment – Designed to test hypothesis Observations – Collect sample measurements Observations – Collect sample measurements Analysis – Do the data formally tend to support or refute the hypothesis, and with what strength? (Lots of juicy formulas...) Analysis – Do the data formally tend to support or refute the hypothesis, and with what strength? (Lots of juicy formulas...) Conclusion – Reject or retain hypothesis; is the result statistically significant? Conclusion – Reject or retain hypothesis; is the result statistically significant? Interpretation – Translate findings in context! Interpretation – Translate findings in context! Statistics is implemented in each step of the classical scientific method! 11

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