USC3002 Picturing the World Through Mathematics

1. USC3002 Picturing the World Through Mathematics Wayne Lawton Department of Mathematics S14-04-04, 65162749 matwml@nus.edu.sg Theme for Semester I, 2008/09 : The Logic of Evolution, Mathematical Models of Adaptation from Darwin to Dawkins

2. PLAN FOR LECTURE • Populations and Samples • 2. Sample Population Statistics • 3. Statistical Hypothesis • 4. Test Statistics for Gaussian Hypotheses • Sample Mean for Parameter Estimation • z-Test and t-Test Statistics • Rejection/Critical Regionforz-Test Statistic • Hypothesis Test for Mean Height • 5. General Hypotheses Tests • Type I and Type II Errrors • Null and Alternative Hypotheses • 6. Assign Tutorial Problems

3. POPULATIONS AND SAMPLES Population - a specified collection of quantities: e.g. heights of males in a country, glucose levels of a collection of blood samples, batch yields of an industrial compound for a chemical plant over a specified time with and without the use of a catalyst Sample Population – a population from which samples are taken to be used for statistical inference Sample - the subset of the sample population consisting of the samples that are taken.

4. SAMPLE POPULATION PARAMETERS Sample Sample Parameters Sample Size Sample Mean Sample Variance Sample Standard Deviation

5. SAMPLE POPULATION PARAMETERS Theorem 1 The variance of a population is related to its mean and average squared values by Proof Since Why ? Question How can the proof be completed ?

6. STATISTICAL HYPOTHESES are assertions about a population that describe some statistical properties of the population. Typically, statistical hypotheses assert that a population consists of independent samples of a random variable that has a certain type of distribution and some of the parameters that describe this distribution may be specified. For Gaussian distributions there are four possibilities: Neither the mean nor the variance is specified. Only the variance is specified. Only the mean is specified. Both the mean and the variance are specified.

7. TEST STATISTICS for Hypothesis with Gaussian Distributions The sample mean for unknown, known is Gaussian with mean 0 and variance 1/n. Proof (Outline) We let < Y > denote the mean of a random variable Y. Then clearly Independence and Theorem 1 gives where

8. PARAMETER ESTIMATION for Hypothesis with Gaussian Distributions The sample mean for unknown, known can be used to estimate the mean since the estimate error is unbiased and converges in the statistical sense that as

9. MORE TEST STATISTICS for Hypothesis with Gaussian Distributions The One Sample z-Test for known is a Gaussian random variable with mean 0,variance 1. The One Sample t-Test for unknown known, is a t-distributed random variable with n-1 degrees of freedom.

10. z-TEST STATISTIC ALPHAS 0 0.5000 0.2000 0.4207 0.4000 0.3446 0.6000 0.2743 0.8000 0.2119 1.0000 0.1587 1.2000 0.1151 1.4000 0.0808 1.6000 0.0548 1.8000 0.0359 2.0000 0.0228 2.2000 0.0139 2.4000 0.0082 2.6000 0.0047 2.8000 0.0026 3.0000 0.0013 0.0500 1.6449 0.0400 1.7507 0.0300 1.8808 0.0200 2.0537 0.0100 2.3263 0.0050 2.5758 0.0040 2.6521 0.0030 2.7478 0.0020 2.8782 0.0010 3.0902 0.0005 3.2905 0.0004 3.3528 0.0003 3.4316 0.0002 3.5401 0.0001 3.7190 0.0001 3.8906

11. CRITICAL REGION FOR alpha=0.05

12. HEIGHT HISTOGRAMS

13. HYPOTHESIS TEST FOR MEAN HEIGHT You suspect that the height of males in a country has increased due to diet or a Martian conspiracy, you aim to support your Alternative Hypothesis by testing the Null Hypothesis You compute a sample mean using 20 samples then compute If the Null Hypothesis is true the probability that is Question Should the Null Hypothesis be rejected ?

14. GENERAL HYPOTHESES TESTS involve Type I Error: prob rejecting null hypothesis if its true, also called the significance level Type II Error: prob failing to reject null hypothesis if its false, also called the power of a test, requires an Alternative Hypothesis that determines the distribution of the test statistic. and more complicated test statistics, such as the One Sample t-Test statistic, whose distribution is determined even though the distributions of the Gaussian random samples, used to compute it, is not.

15. Homework 5. Due Monday 20.10.08 1. Compute the power of a hypothesis test whose null hypothesis is that in vufoil #13, the alternative hypothesis asserts that heights are normally distributed with where and are the same as for the null hypothesis and 20 samples are used and the significance Suggestion: if the alternative hypothesis is true, what is the distribution of test statistic What is the probability that 2. Use a t-statistic table to describe how to test the null hypothesis that heights are normal with mean and unknown variance based on 20 samples.

16. EXTRA TOPIC: CONFIDENCE INTERVALS Given a sample mean for large we can assume, by the central limit theorem that it is Gaussian with mean mean of the original population and variance of the original population. variance Furthermore, sample variance and if the population is {0,1}-valued We say that with confidence where p(x) is the probability density of a Gaussian with mean and standard deviation is a random variable unif. on [-L,L] Theorem If then Bayes Theorem 

17. EXTRA TOPIC: TWO SAMPLE TESTS A null hypothesis may assert a that two populations have the same means, a special case for {0,1}-valued populations asserts equalily of population proportions. Under these assumptions and if the variances of both populations are known, hypothesis testing uses the Two-Sample z-Test Statistic where is the sample mean, variance, and sample size for one population, tilde’s for the other. For unkown variances and other cases consult: http://en.wikipedia.org/wiki/Statistical_hypothesis_testing

18. EXTRA TOPIC: CHI-SQUARED TESTS are used to determine goodness-or-fit for various distributions. They employ test statistics of the form where are independent observations & null hyp.  expected value and chi-squared distrib. with d-1 degrees of freedom. Example [1,p.216] A geneticist claims that four species of fruit flies should appear in the ratio 1:3:3:9. Suppose that the sample of 4000 flies contained 226, 764, 733, and 2277 flies of each species, respectively. For alpha = .1, is there sufficient evidence to reject the geneticist’s claim ? Answer: The expected values are 250, 750, 750, 2250 hence NO since 3 deg. freed. & alpha = .1

19. EXTRA TOPIC: POISSON APPROXIMATION The Binomial Distribution is the probability that k-events happen in n-trials if It has mean and variance If and then The right side is the Poisson Distribution

20. REFERENCES • Martin Sternstein, Statistics, Barrows College • Review Series, New York, 1996. Survey textbook • covers probability distributions, hypotheses tests, • populations,samples, chi-squared analysis, regression. 2. E. L. Lehmann, Testing Statistical Hypotheses, New York, 1959. Detailed development of the Neyman-Pearson theory of hypotheses testing. 3. J.Neyman and E.S. Pearson, Joint Statistical Papers, Cambridge University Press, 1967. Source materials. 4. Jan von Plato, Creating Modern Probability, Cambridge University Press, 1994. Charts the history and development of modern probability theory.