.. title: Engineering Probability Class 25 Mon 2019-04-15 .. slug: class25 .. date: 2019-04-15 .. tags: mathjax .. category: class .. link: .. description: .. type: text .. sectnum:: .. contents:: Table of contents .. Iclicker -------- All these questions concern math SAT scores, which we assume have a mean of 500 and standard deviation of 100. #. What is the probability that one particular score is between 400 and 600? A. .34 #. .68 #. .96 #. .98 #. .9974 #. I take a random sample of 4 students, and compute the mean of their 4 scores. What is the probability that that mean is between 400 and 600? A. .34 #. .68 #. .96 #. .98 #. .9974 #. I take a random sample of 9 students, and compute the mean of their 9 scores. What is the probability that that mean is between 400 and 600? A. .34 #. .68 #. .96 #. .98 #. .9974 #. What is the standard deviation of the 4-student sample? A. 25 #. 33 #. 50 #. 100 #. 200 #. What is the standard deviation of the 9-student sample? A. 25 #. 33 #. 50 #. 100 #. 200 Statistics ---------- Now, we learn **statistics**. That means, determining parameters of a population by sampling it. In **probability**, we already know that parameters, and calculate things from them. We'll start with Leon-Garcia Chapter 8, and add stuff to it. This course module fits with RPI's goal of a data dexterity requirement for undergrads. Pres Jackson mentioned this at the spring town meet; see https://president.rpi.edu/speeches/2019/remarks-spring-town-meeting . Disclosure: Prof Dave Mendonca and I are co-chairs of the SoE Data Dexterity Task Force, working out details of this. Hypothesis testing, from text (plus extras) ====================================================== #. Say we want to test whether the average height of an RPI student (called the population) is 2m. #. We assume that the distribution is Gaussian (normal) and that the standard deviation of heights is, say, 0.2m. #. However we don't know the mean. #. We do an experiment and measure the heights of n=100 random students. Their mean height is, say, 1.9m. #. The question on the table is, is the population mean 2m? #. This is different from the earlier question that we analyzed, which was this: What is the most likely population mean? (Answer: 1.9m.) #. Now we have a hypothesis (that the population mean is 2m) that we're testing. #. The standard way that this is handled is as follows. #. Define a null hypothesis, called H0, that the population mean is 2m. #. Define an alternate hypothesis, called HA, that the population mean is not 2m. #. Note that we observed our sample mean to be $0.5 \\sigma$ below the population mean, if H0 is true. #. Each time we rerun the experiment (measure 100 students) we'll observe a different number. #. We compute the probability that, if H0 is true, our sample mean would be this far from 2m. #. Depending on what our underlying model of students is, we might use a 1-tail or a 2-tail probability. #. Perhaps we think that the population mean might be less than 2m but it's not going to be more. Then a 1-tail distribution makes sense. #. That is, our assumptions affect the results. #. The probability is Q(5), which is very small. #. Therefore we reject H0 and accept HA. #. We make a type-1 error if we reject H0 and it was really true. See http://en.wikipedia.org/wiki/Type_I_and_type_II_errors #. We make a type-2 error if we accept H0 and it was really false. #. These two errors trade off: by reducing the probability of one we increase the probability of the other, for a given sample size. #. E.g. in a criminal trial we prefer that a guilty person go free to having an innocent person convicted. #. Rejecting H0 says nothing about what the population mean really is, just that it's not likely 2m. #. Enrichment. Random sampling is hard. The US government got it wrong here: http://politics.slashdot.org/story/11/05/13/2249256/Algorithm-Glitch-Voids-Outcome-of-US-Green-Card-Lottery #. The above tests, called **z-tests**, assumed that we know the population variance. #. If we don't know the population variance, we can estimate it by sampling. #. We can combine estimating the population variance with testing the hypothesis into one test, called the **t-test**. Dr Nic's videos =============== Understanding the Central Limit Theorem https://www.youtube.com/watch?v=_YOr_yYPytM Variation and Sampling Error https://www.youtube.com/watch?v=y3A0lUkpAko Understanding Statistical Inference https://www.youtube.com/watch?v=tFRXsngz4UQ Understanding Hypothesis testing, p-value, t-test - Statistics Help https://www.youtube.com/watch?v=0zZYBALbZgg Research By Design videos ========================= 10-1 Guinness, Student, and the History of t Tests https://www.youtube.com/watch?v=bqfcFCjaE1c