two sample exact binomial test
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The p-value is the area of the red bars. Note. [NOISE] [NOISE] [NOISE] Okay, so let's put some context on this. $\rho = $ “the true proportion of children in the population who play baseball”. Other types of claims and sampling methods will have different mathematical models and involve different types of distributions. and then if we were doing a two-sided test remember that we would double this P value. $\text{(claim) } \ H_A: \rho < 49\%$ $= 0.04031078 + 0.006046618$ However, when we surveyed 200 children we found that only 50% said they liked chocolate. And that's, that, that turns out to be a good thing to do, because the binomial confidence interval you don't want it centered exactly at p hat, because the binomial distribution is p hat is Further away from one half gets more asymmetric. And we can, you know, of course we can use the test statistic to just perform the test, so I should have said this on the previous slide. And what we'd like to do is to, to do a test of whether or not the side effects are the propensity for side effects is the same within the two, within the two drugs. But it's good, I think, whenever you're doing these. The p-value is the area of the red bars. $+ 0.0188641767342666$ Are all people who are professional drug takers for pharmaceutical companies? We will take the table which $+ 200C200\ (51\%)^{200}\ (49\%)^{0}$ It the authors wanted to include this contrary result, the authors might write something like: We had thought that more than 51% of children liked chocolate. Since the bars are very thin, if we put numbers on top of them, the result will be a mess. The p-value method of hypothesis testing is probably the most common way to test a claim. The One Sample Proportion Test is used to estimate the proportion of a population. $X = $ the random variable that counts how many of the students in such samples are stem majors. think about what that means in the context of the problem that you're studying. So just give you an example lets assume that our sample of people. Since $n = 10$ we have: Video showing how to solve Question 1 above (9:35): Continuation of above video showing solution to Question 2 below (3:34). They will often state the claim as if it were a fact (even though it isn’t). which is usually not the case. They probably adhere to their medication schedule very very precisely and, and other things like that, they're probably very good takers of medications. Figure for Question 4. Answer to Question 1. $X(\text{test-sample}) = 8$ so samples that provide as much support are samples which satisfy $X = 8, 9$, or $10$. And the alpha quantile if you're doing in one sided test well where you're testing p less than p0. So, for example $P(X=8) = 0.121$. If the data from the sample supports the claim we can conduct a formal hypothesis test to determine if the sample provides statistically significant evidence that the claim is true. Then read through the first two or three Questions and their solutions. Here is the R script used to create the above barplot. $n = $ the size of our sample, so $n = 12$. Recall the binomial distribution formula: $$P(X = r) = nCr\ \rho^r \ (1 – \rho)^{n-r}$$. so mechanically I, I hope you find this easy, but let's talk a little bit about the thinking of this. • Bernoulli trial (or binomial trial) - a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted • For testing, the binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement Either way, 6.7's going to be bigger than it. Unfortunately, if we try to minimize the chances of making a type I error, we need to be more cynical, and that will increase the likelihood of making a type II error. $+ 10C9\ (60\%)^9 \ (40\%)^{1} $ The p-value is the area of the red bars. if n is small then, then, then this term in front of the one, this term in front of the one half gets a little bit bigger and, well, the one half probably hopefully doesn't dominate but, but, but there's more a greater fraction placed on the, on the one half. Notation. To save money, we could just make up the data and calculate the p-value; or we could collect the data in a sloppy manner, or make a mess of the data in a million ways; but what would a p-value based on such data tell us about the real world? A type II error is when a true claim is rejected as false 1. Claim: More than 60% of students are STEM majors. $+ 12C1\ (49\%)^1 \ (51\%)^{11}$ “as much support” means the same or stronger support. That it's more likely for anyone else around them to get side effects who also received the drug. $\rho = $ “the true proportion of students in the population who are STEM majors”. Here is the probability density barplot corresponding to the p-value calculated in Question 1, above: Figure for Question 1. A small sample size will often result in a large p-value. $X(\text{our sample}) = 3$. $+ 200C116\ (51\%)^{116} \ (49\%)^{84}$ Okay, so our obvious metric of the discrepancy between p hat and p naught would be the difference or maybe we can do a long ratio or something like that but lets for right now lets do the difference. If the sample provided statistical significance, like in Question 6, the authors might write something like: More than 51% of children like chocolate (p = .0383, exact binomial test). $X = $ the random variable that counts how many of the students in such samples are stem majors. So here we have a table where there's 40 total people. Claim: Less than 49% of children play baseball. So they are they are two proportions that add up that, that, that, um,add up to one. Serve as motivation for creating a confidence interval, as well.
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