Entity Academy Lesson 6 Statistical Inference

Sindy Saintclair Monday, November 28 2021 Lesson 6 – Statistical Inference Learning Objectives and Questions Notes and Answers Sampling Methods Balance Accuracy with Practicality Sampling – when you take a subset of the population and make assertions about the entire population by just observing a small subset of that population. - population: everyone (not as practical) - sample: some (more practical)  The risk involved with this is since you are purposely excluding most of the households in the city, you are obtaining incomplete information. You can take multiple samples and they will all be slightly different, with no way to tell which one is the most accurate.  The advantage is that my workload is dramatically decreased if I decide to sample. Sample Size Sample size is referred to as n. For instance, if you talk to 30 people out of a larger group, this is a sample size of 30, or n=30. Simple Random Sample Everyone has an equal chance of being selected - drawing names out of hat - scientists will assign people a number and will have Excel or Python select someone Cluster Sampling Randomly select a group, not an individual Example Stratified Sampling – usually demographic in nature Population 10,000 Sample: 100 Females: 20% - 2,000 Female: 20% - 20 Males: 80% - 8,000 Males: 80% Systematic Sampling

Convenience Sampling Just sampling those that are easy to sample, for instance only the people closest to you, the smallest file, or the beginning of the alphabet— usually what’s convenient for the scientist. Sample Size: Number of People in the Sample - How many people do you need in your sample? Enough to represent the population accuracy Not too many to be impractical Simple Random Sampling Often referred to as SRS, every potential candidate for data collection has the same probability of being chosen as every other candidate for data collection. Whoever gets selected is random . The best way to do this is to assign all candidates a unique number, and then have a random number generator select which of those candidates should be a part of the sample. This method provides the best samples but may not always be done because it can be logistically difficult or expensive. Sampling Method Examples I work for a healthcare provider. My team is tasked by a federal agency to add to the knowledge base of 10 th grade students by collecting medical information. The population is all 10 th grade students in the state of Ohio. You will collect height, weight, and hearing test data for each child sampled. Simple Random Sampling Example Each 10 th grader in Ohio is assigned a number from 1 to 38,559 (that is the number of 10 th graders in

sampling, you don’t randomly select the strata and sample everyone in it. Instead, you will randomly sample within the strata at a rate that is proportionate to the how large the strata are. In this image, since the purple strata is the largest, you will randomly select more people from those strata then from either the red or the green strata, which are smaller. Example – There are some huge high schools in Ohio (3500+) students, and there are some smaller schools and charter schools with as few as 50 students. I break the schools into three different strata: schools with 1500 or more students, schools with 400 to 1499 students, and schools with 399 or fewer students. I will label these schools as big, medium, and small schools. The big schools are where 45% of Ohio 10 th grade students are going to school, with 38% going to medium-sized schools, and 17% going to the small schools. I want my total sample size to be 3000 students, so I randomly select 3000 x 45% = 1350 students from the big schools ,3000 x 38% = 1140 students from the medium schools, and 3000 x 17% = 510 students from the small schools. This way, I have broken down the population into stratum and sampled from each strata proportionally so that each is fairly represented in the final sample of 3000 students. Systematic Sampling Systematic sampling happens when a start point is chosen at random, and then every nth item is selected after that. This ultimately makes the sampling as a whole no longer random. Example

Each 10 th grader is assigned a number again, and then a random number between 1 and 13 is generated by a random number generator. Suppose you generate the number 10. Then, every 13 th student after that is selected. In this case, that would be student number 10, 23, 36, 49, 62, etc. Sampling in this manner will get you pretty close to sample size of 3000. Convenience Sampling When someone makes little or no effort to randomly sample, and they collect the information that is easiest to get. Convenience sampling is usually the most biased type of sampling, but unfortunately, the most common. Example Right across the street from your facility is a high school. You send a team member over there to measure all of the 10 th graders at that school. One of your team members has a spouse that teaches high school, so tomorrow she goes with her husband to the high school and measures all of those 10 th – grade students there. Another team member has a cousin who is the school nurse at a high school in another city, so you asked the cousin to collect data from the 10 th grade students at that high school, and so forth. It turns out that for some of the high schools, there was a volleyball tournament, so some of the 10 th graders were missing because they are at the tournament, and another high school had a bunch of students missing because of band camp, but you just don’t care. Choosing a Dependent on one’s sampling goals and what is the

population anyway - because you want to know something about a bigger set, usually called a population. - helps me to develop estimates of what is going on with the whole population, and usually can provide you with a confidence interval around that estimate - Balance rigor and feasibility Why Analyze Data? From a practicality perspective: - is 1.1 really bigger than 1? - is 2.5 really bigger than 2.3? Hypothesis Testing Statistical inference has been called the science of comparison. Mathematics has a lot of branches, and most of them are academic and theoretical. Null : H 0  equal Alternative : H a or H 1  not equal, greater than, or lesser than Steps of Hypothesis Testing - create a hypothesis - calculate a test statistic - calculate the probability – p value - determine alpha - decision rule Determine Alpha (α) α Level Accuracy Error α = 0.01 99% chance of accuracy 1% chance of error α = 0.05 95% chance of accuracy 5% chance of error α = 0.10 90% chance of accuracy 10% chance of error Alpha is a pre-determined number that represents how often you are willing to draw a conclusion from the data but be wrong. I will compare my p value with the alpha value to determine whether something is significant or not. The most typical value for alpha is 0.05. This means that there is a 95% chance that my results are accurate, and I am willing to accept a 5% chance that my data is wrong. However, on the more rigorous side, I may also see alpha = 0.01, Rigor

meaning that there is a 99% chance that my results are accurate, and I am willing to accept a 1% chance of error. And on the less rigorous side, from time to time I may see an alpha of 0.10, meaning that there is a 90% chance my results are accurate and a 10% chance my data is inaccurate. Creating a Hypothesis Statistical inference is used to test hypotheses. But to test a hypothesis, it has to be created. The most important part of creating a hypothesis is to remember that it must contain the ‘=’ sign, either explicitly or implicitly. Example #1) You have invented a new drug to treat HTN. You are sure your drug is better than the most commonly used current drug, but as you develop a hypothesis, you hypothesize that the two drugs are exactly the same in effectiveness. What you really want to do is prove yours is better but will start by assuming they are both equally good. H 0 : new drug = baseline drug H α : new drug > baseline drug Example #2) You are taking a survey and you believe that the crab cakes sold at “The Shack” are better than the crab cakes sold at “Seafood Supreme.” Your hypothesis should be that both crab cakes are equally desirable to customers, and you hope to use data to disprove the hypothesis. H 0 : The shack = Seafood Supreme H α : The shack > Seafood Supreme Example #3) You want to compare baseball players from 2 different eras, and you think one is better than the other. Your hypothesis is that they are equally skilled, and then you hope to use data to disprove that hypothesis. H 0 : Player A = Player B H α : Player A ≠ Player B Example #4) You want to collect some data to see if movie preference is dependent on gender. Your

A: Accept F: False EXAMPLES COURTROOM ANALOGY The null hypothesis is not guilty because everyone is innocent until proven guilty. if the null is false, then the person will get arrested if I reject the null. If I chose to accept the null, that would be a type II error. if the null is true, then the person goes free if I accept the null. If I chose to reject the null, that would be a type I error. Type II Error Example – if the defendant is guilty, but the jury finds him “not guilty”, they have committed a Type II error. Type I Error Example – if the defendant did not commit the crime, but the jury finds him guilty anyway, the jury has committed a Type I Error. A Real-Life Example of Type II Error OJ Simpson murder trial – in short, the test doesn’t determine whether or not A=B, it determines if there is enough convincing evidence that A and B are different, or not just as trials don’t determine innocence or guilt but whether if the prosecution has enough convincing evidence to find the defendant guilty, or insufficient evidence, which will lead to a verdict of being not guilty.