# Hypothesis Testing for Beginners

Size: px
Start display at page:

Transcription

1 Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

2 One year ago a friend asked me to put down some easy-to-read notes on hypothesis testing. Being a student of Osteopathy, he is unfamiliar with basic expressions like random variables or probability density functions. Nevertheless, the profession expects him to know the basics of hypothesis testing. These notes offer a very simplified explanation of the topic. The ambition is to get the ideas through the mind of someone whose knowledge of statistics is limited to the fact that a probability cannot be bigger than one. People familiar with the topic will find the approach too easy and not rigorous. But this is fine, these notes are not intended to them. For comments, typos and mistakes please contact me on m.piffer@lse.ac.uk Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

3 Plan for these notes Describing a random variable Expected value and variance Probability density function Normal distribution Reading the table of the standard normal Hypothesis testing on the mean The basic intuition Level of significance, p-value and power of a test An example Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

4 Random Variable: Definition The first step to understand hypothesis testing is to understand what we mean by random variable A random variable is a variable whose realization is determined by chance You can think of it as something intrinsically random, or as something that we don t understand completely and that we call random Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

5 Random Variable: Example 1 What is the number that will show up when rolling a dice? We don t know what it is ex-ante (i.e. before we roll the dice). We only know that numbers from 1 to 6 are equally likely, and that other numbers are impossible. Of course, we would not consider it random if we could keep track of all the factors affecting the dice (the density of air, the precise shape and weight of the hand...). Being impossible, we refer to this event as random. In this case the random variable is {Number that appears when rolling a dice once} and the possible realizations are {1, 2, 3, 4, 5, 6 } Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

6 Random Variable: Example 2 How many people are swimming in the lake of Lausanna at 4 pm? If you could count them it would be, say, 21 on June 1st, 27 on June 15th, 311 on July 1st... Again, we would not consider it random if we could keep track of all the factors leading people to swim (number of tourists in the area, temperature, weather..). This is not feasible, so we call it a random event. In this case the random variable is {Number of people swimming in the lake of Lausanne at 4 pm} and the possible realizations are {0, 1, 2,... } Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

7 Moments of a Random Variable The world is stochastic, and this means that it is full of random variables. The challenge is to come up with methods for describing their behavior. One possible description of a random variable is offered by the expected value. It is defined as the sum of all possible realizations weighted by their probabilities In the example of the dice, the expected value is 3.5, which you obtain from E[x] = = 3, 5 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

8 Moments of a Random Variable The expected value is similar to an average, but with an important difference: the average is something computed ex-post (when you already have different realizations), while the expected value is ex-ante (before you have realizations). Suppose you roll the dice 3 times and obtain {1, 3, 5}. In this case the average is 3, although the expected value is 3,5. The variable is random, so if you roll the dice again you will probably get different numbers. Suppose you roll the dice again 3 times and obtain {3, 4, 5}. Now the average is 4, but the expected value is still 3,5. The more observations extracted and the closer the average to the expected value Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

9 Moments of a Random Variable The expected value gives an idea of the most likely realization. We might also want a measure of the volatility of the random variable around its expected value. This is given by the variance The variance is computed as the expected value of the squares of the distances of each possible realization from the expected value In our example the variance is 2,9155. In fact, x probability x-e(x) (x-e(x))^2 prob*(x-e(x))^2 1 0,1666-2,5 6,25 1, ,1666-1,5 2,25 0, ,1666-0,5 0,25 0, ,1666 0,5 0,25 0, ,1666 1,5 2,25 0, ,1666 2,5 6,25 1, ,9155 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

10 Moments of a Random Variable Remember, our goal for the moment is to find possible tools for describing the behavior of a random variable Expected values and variances are two, fairly intuitive possible tools. They are called moments of a random variable More precisely, they are the first and the second moment of a random variable (there are many more, but we don t need them here) For our example above, we have First moment = E(X) = 3,5 Second moment = Var(X) = 2,9155 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

11 PDF of a Random Variable The moments are only one possible tool used to describe the behavior of a random variable. The second tool is the probability density function A probability density function (pdf) is a function that covers an area representing the probability of realizations of the underlying values Understanding a pdf is all we need to understand hypothesis testing Pdfs are more intuitive with continuous random variables instead of discrete ones (as from example 1 and 2 above). Let s move now to continuous variables Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

12 PDF of a Random Variable: Example 3 Consider an extension of example 1: the realizations of X can still go from 1 to 6 with equal probability, but all intermediate values are possible as well, not only the integers {1, 2, 3, 4, 5, 6 } Given this new (continuous) random variable, what is the probability that the realization of X will be between 1 and 3,5? What is the probability that it will be between 5 and 6? Graphically, they are the area under the pdf in the segments [1; 3,5] and [5; 6] Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

13 PDF of a Random Variable The graphic intuition is simple, but of course to compute these probabilities we need to know the value of the parameter c, i.e. how high is the pdf of this random variable To answer this question, you should first ask yourself a preliminary question: what is the probability that the realization will be between 1 and 6? Of course this probability is 1, as by construction the realization cannot be anywhere else. This means that the whole area under a pdf must be equal to one In our case this means that (6-1)*c must be equal to 1, which implies c = 1/5 = 0,2. Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

14 PDF of a Random Variable Now, ask yourself another question: what is the probability that the realization of X will be between 8 and 12? This probability is zero, given that [8, 12] is outside [1,6] This means that above the segment [8,12] (and everywhere outside the support [1, 6]) the area under the pdf must be zero. To sum up, the full pdf of this specific random variable is Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

15 PDF of a Random Variable In our example we see that: Prob(1 < X < 3, 5) = (3, 5 1) 1/5 = 1/2 Prob(5 < X < 6) = (6 5) 1/5 = 1/5 Prob(8 < X < 24) = (24 8) 0 = 0 Hypothesis testing will rely extensively on the idea that, having a pdf, one can compute the probability of all the corresponding events. Make sure you understand this point before going ahead Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

16 Normal Distribution We have seen that the pdf of a random variable synthesizes all the probabilities of realization of the underlying events Different random variables have different distributions, which imply different pdfs. For instance, the variable seen above is uniformly distributed in the support [1,6]. As for all uniform distributions, the pdf is simply a constant (in our case 0,2) Let s introduce now the most famous distribution, which we will use extensively when doing hypothesis testing: the normal distribution Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

17 Normal Distribution The normal distribution has the characteristic bell-shaped pdf: Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

18 Normal Distribution Contrary to the uniform distribution, a normally distributed random variable can have realizations from to +, although realizations in the tail are really rare (the area in the tail is very small) The entire distribution is characterized by the first two moments of the variable: µ and σ 2. Having these two moments one can obtain the precise position of the pdf When a random variable is normally distributed with expected value µ and variance σ 2, then it is written We will see this notation again X N(µ, σ 2 ) Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

19 Normal Distribution Let s consider an example. Suppose we have a normally distributed random variable with µ = 8 and σ 2 = 16. What is the probability that the realization will be below 7? What is the probability that it will be in the interval [8, 10]? Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

20 Standard Normal Distribution Graphically this is easy. But how do we compute them? To answer this question we first need to introduce a special case of the normal distribution: the standard normal distribution The standard normal distribution is the distribution of a normal variable with expected value equal to zero and variance equal to 1. It is expressed by the variable Z: Z N(0, 1) The pdf of the standard normal looks identical to the pdf of the normal variable, except that it has center of gravity at zero and has a width that is adjusted to allow the variance to be one Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

21 Standard Normal Distribution What is the big deal of the standard normal distribution? It is the fact that we have a table showing, for each point in [, + ], the probability that we have to the left and to the right of that point. For instance, one might want to know the probability that a standard normal has a realization lower that point From these table one gets that it is 0,99. What is then the probability that the realization is above 2,33? 0,01, of course. Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

22 Standard Normal Distribution The following slide shows the table for the standard normal distribution. On the vertical and horizontal axes you read the z-point (respectively the tenth and the hundredth), while inside the box you read the corresponding probability The blue box shows the point we used to compute the probability in the previous slide Make sure you can answer the following questions before going ahead Question 1: what is the probability that the standard normal will give a realization below 1? Question 2: what is the point z below which the standard normal has 0,95 probability to occur? Question 3: what is the probability that the standard normal will give a realization above (not below) -0.5 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

23 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

24 Standard Normal Distribution Answers to our previous questions 1) Prob(Z < 1) = 0, ) z so that Prob(Z < z) = 0, 95 is 1, 64 3) 1 Prob(Z < 0.5) = 0, 6915 Last, make sure you understand the following theorem: if X is normally distributed with expected value µ and variance σ 2, then if you subtract µ from X and divide everything by σ 2 = σ you obtain a new variable, which is a standard normal: Given X N(µ, σ 2 ) X µ σ N(0, 1) = Z Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

25 Normal Distribution We are finally able to answer our question from a few slides before: what is the value of Prob(X < 7) and Prob(8 < X < 10) given X N(8, 16)? Do we have a table for this special normal variable with E(x)=8 and Var(x)=16? No! But we don t need it: we can do a transformation to X and exploit the table of the standard normal Remembering that you can add/subtract or multiply/divide both sides of an inequality, we obtain.. Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

26 Normal Distribution Prob(X < 7) = Prob(X 8 < 7 8) = Prob(X 8 < 1) = = Prob( X 8 4 < 1 4 ) = Prob(X 8 4 < 0, 25) The theorem tells us that the variable X µ σ = X 8 4 is a standard normal, which greatly simplifies our question to: what is the probability that a standard normal has a realization on the left of point 0, 25? We know how to answer this question. From the table we get Prob(X < 7) = Prob(Z < 0, 25) = 1 Prob(Z > 0.25) = 1 0, 5987 = = 40, 13% Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

27 Normal Distribution Let s now compute the other probability: Prob(8 < X < 10) = Prob(X < 10) Prob(X < 8) = = Prob(Z < 10 8 ) Prob(Z < ) = Prob(Z < 0.5) Prob(Z < 0) = 0, = = 0, 1915 = 19, 15% To sum up, a normal distribution with expected value 8 and variance 16 could have realizations from to. But we now know that the probability that X will be lower than 7 is 40,13 %, while the probability that it will be in the interval [8,10] is 19,15 % Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

28 Normal Distribution Graphically, given X N(8, 16) Exercise: show that Prob(X > 11) = 22, 66% Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

29 Hypothesis Testing We are finally able to use what we have seen so far to do hypothesis testing. What is this about? In all the exercises above, we assumed to know the parameter values and investigated the properties of the distribution (i.e. we knew that µ = 8 and σ 2 = 16). Of course, knowing the true values of the parameters is not a straightforward task By inference we mean research of the values of the parameters given some realizations of the variable (i.e. given some data) Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

30 Hypothesis Testing There are two ways to proceed. The first one is to use the data to estimate the parameters, the second is to guess a value for the parameters and ask the data whether this value is true. The former approach is estimation, the latter is hypothesis testing In the rest of the notes we will do hypothesis testing on the expected value of a normal distribution, assuming that the true variance is known. There are of course tests on the variance, as well as all kinds of tests Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

31 Hypothesis Testing Consider the following scenario. We have a normal random variable with expected value µ unknown and variance equal to 16 We are told that the true value for µ is either 8 or 25. For some external reason we are inclined to think that µ = 8, but of course we don t really know for sure. What we have is one realization of the random variable. The point is, what is this information telling us? Of course, the closer is the realization to 8 and the more I would be willing to conclude that the true value is 8. Similarly, the closer the realization is to 25 and the more I would reject that the hypothesis that µ = 8 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

32 Hypothesis Testing How do we use the data to derive some meaningful procedure for inferring whether my data is supporting my null hypothesis of µ = 8 or is rejecting it in favor of the alternative µ = 25? Let s first put things formally: X N(µ, 16), with H 0 : µ = 8 vs. H a : µ = 25 Call x the realization of X which we have and which we use to do inference. What is the distribution that has generated this point, X N(8, 16) or X N(25, 16)? Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

33 Hypothesis Testing If null hypothesis is true, the realization x comes from If alternative hypothesis is true, the realization x comes from Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

34 Hypothesis Testing The key intuition is: how far away from 8 must the realization be for us to reject the null hypothesis that the true value of µ is 8? Suppose x = 12. In this case x is much more likely to come from X N(8, 16) rather than from X N(25, 16). Similarly, if x = 55 then this value is much more likely to come from X N(25, 16) rather than from X N(8, 16) The procedure is to choose a point c (called critical value) and then reject H 0 if x > c, while not reject H 0 if x < c. The point is to find this c in a meaningful way Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

35 Hypothesis Testing Start with the following exercise. For a given c, what is the probability that we reject H 0 by mistake, i.e., when the true value for µ was actually 8? As we have seen, this is nothing but thee area under X N(8, 16) on the interval [c, + ] Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

36 Hypothesis Testing Given c, we also know how to compute this probability. Under H 0, it is simply Prob(X > c) = Prob(Z > c 8 4 ) What we just did is: given c, compute the probability of rejecting H 0 by mistake. The problem is that we don t have c yet, that s what we need to compute! A sensible way to choosing c is to do the other way around: find c so that the probability of rejecting H 0 by mistake is a given number, which we choose at the beginning. In other words, you choose with which probability of rejecting H 0 by mistake you want to work and then you compute the corresponding critical value Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

37 Hypothesis Testing In jargon, the probability of rejecting H 0 by mistake is called type 1 error, or level of significance. It is expressed with the letter α Our problem of finding a meaningful critical value c has been solved: at first, choose a confidence interval. Then find the point c corresponding to the interval α. Having c, one only needs to compare it to the realization x from the data. If x > c, then we reject H 0 = 8, knowing that we might be wrong in doing so with probability α The most frequent values used for α are 0.01, 0,05 and 0,1 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

38 Hypothesis Testing Let s do this with our example. Suppose we want to work with a level of significance of 1 % The first thing to do is to find the critical vale c which, if H 0 was true, would have 1 % probability on the right. Formally, find c so that 0, 01 = Prob(X > c) = Prob(Z > c 8 4 ) What is the point of the standard normal distribution which has area of 0,01 on the right? This is the first value we saw when we introduced the N(0,1) distribution. The value was 2,33: 0, 01 = Prob(Z > c 8 ) = Prob(Z > 2, 33) 4 Impose c 8 4 = 2, 33 and obtain c = 17, 32. Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

39 Hypothesis Testing At this point we have all we need to run our test. Suppose that the realization from X that we get is 15. Is x =15 far enough from µ = 8 to lead us to reject H 0? Clearly 15 < 17, 32, which means that we cannot reject the hypothesis that µ = 8 Suppose we run the test with another realization of X, say x = 20. Is 20 sufficiently close to µ = 25 to lead us to reject H 0? Of course it is, given that 20 is above the critical c = 17, 32 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

40 Hypothesis Testing Note that when rejecting H 0 we cannot be sure at 100% that the true value for the parameter µ is not 8. Similarly, when we fail to reject H 0 we cannot be sure at 100 % that the true value for µ is actually 8. What we know is only that, in doing the test several times, we would mistakenly reject µ = 8 with α = 0.01 probability. Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

41 Hypothesis Testing Our critical value c would have been different, had we chosen to work with a different level of significance. For instance, the critical value for α = 0, 05 is 14,56, while the critical value for α = 0, 1 is 13,12 (make sure you know how to compute these values) α c Suppose that the realization of X is 15. Do we reject H 0 when working with a level of significance of 1 %? No. But do we reject it if we work with a level of significance of 10 %? Yes! The higher the probability that we accept in rejecting H 0 by mistake and the more likely it is that we reject H 0 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

42 Hypothesis Testing Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

43 Hypothesis Testing If, given x, the outcome of the test depends on the level of significance chosen, can we find a probability p so that we will reject H 0 for all levels of significance from 100 % down to p? To see why this question is interesting, compute the probability p on the right of x = 15. We will be able to reject H 0 for all critical values lower than x, or equivalently, for all levels of significance higher than p p = Prob(X > x ) = Prob(Z > 15 8 ) = Prob(Z > 1, 75) = 4 = 1 Prob(Z < 1, 75) = 0, 0401 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

44 Hypothesis Testing Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

45 Hypothesis Testing If we have chosen α < p, then it means that x < c and we cannot reject H 0 If we have chosen an α > p, then it means that x > c and we can reject H 0 In our example, we can reject H 0 for all levels of significance α 4, 01% The probability p is called p-value. The p-value is the lowest level of significance which allows to reject H 0. The lower p and the easier will be to reject H 0, since we will be able to reject H 0 with lower probabilities of rejecting it by mistake. In other words, the lower p and the more confident you are in rejecting H 0 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

46 Hypothesis Testing Our last step involves the following question. What if the true value for µ was actually 25? What is the probability that we mistakenly fail to reject H 0 when the true µ was actually 25? Given what we saw so far this should be easy: it is the area on the region [, c] under the pdf of the alternative hypothesis: Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

47 Hypothesis Testing In jargon, the probability of not rejecting H 0 when it is actually wrong is called type 2 error. It is expressed by the letter β Let s go back to our example. At this point you should be able to compute β easily. Under H a we have β = Prob(X > c) = Prob(X > 17, 32) = = Prob( X 25 4 < 17, ) = Prob(Z < 1, 92) = 0, Last, what is the probability that we reject H 0 when it was actually wrong and the true one was H a? It is obviously the area under X N(25, 16) in the interval [c, + ], or equivalently, 1 β = 0, This is the power of the test Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

48 Hypothesis Testing The power of a test is the probability that we correctly reject the null hypothesis. It is expressed by the letter π Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

49 Hypothesis Testing To sum up, the procedure for running an hypothesis testing requires to: Choose a level of significance α Compute the corresponding critical value and define the critical region for rejecting or not rejecting the null hypothesis Obtain the realization of your random variable and compare it to c If you reject H0, then you know that the probability that you are wrong in doing so is α Given H 1 and the level c, compute the power of the test If you reject the H0 in favor of H 1, then you know that you are right in doing so with probability given by π Alternatively, one can compute the p-value corresponding to the realization of X given by the data and infer all levels of significance up to which you can reject H 0 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

50 Hypothesis Testing In the example seen so far, choosing α = 0, 01, we reject H 0 for all x > 17, 32 and do not reject it for all x < 17, 32 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

51 Hypothesis Testing To conclude, three remarks are necessary: Remark 1: What if we specify that the alternative hypothesis is not a single point, but it is simply H a : µ 8? Such a test is called two-sided test, compared to the one-sided test seen so far. Nothing much changes, other than the fact that the rejection region will be divided into three parts. Find c 1.c 2 so that α = Prob(X < c 1 or X > c 2 ) = 1 Prob(c 1 < X < c 2 ) Then, you reject H 0 if x lies outside the interval [c 1, c 2 ] and do not reject H 0 otherwise Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

52 Hypothesis Testing Remark 2: why should one use only a single realization of X? Doesn t it make more sense to extract n observations and then compute the average? Yes, of course. Nothing much changes in the interpretation. The only note of caution is on the distribution used. It is possible to show that, if X N(µ, σ 2 ), then with X = 1 n n i=1 X i X N(µ, σ2 n ) The construction of the test follows the same steps, but you have to use the new distribution Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

53 Hypothesis Testing Remark 3: so far we assumed that we knew the true value of σ 2. But what is it? Shouldn t we estimate it? Doesn t this change something? Yes, or course. One can show that the best thing to do is to substitute σ 2 with the following statistic: s 2 = 1 n 1 n (X 1 X ) 2 The important difference in this case is that the statistics will not follow the normal distribution, but a different distribution (known as the t distribution). The logical steps for constructing the test are the same. For the details, have a look at a book of statistics i=1 Michele Piffer (LSE) Hypothesis Testing for Beginners August, / 53

### Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

### The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

### Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### Confidence Intervals for One Standard Deviation Using Standard Deviation

Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

### Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide

### Two-sample inference: Continuous data

Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

### 1 Another method of estimation: least squares

1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

### Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

### Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

### Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

### Independent samples t-test. Dr. Tom Pierce Radford University

Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of

### Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes

Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes Simcha Pollack, Ph.D. St. John s University Tobin College of Business Queens, NY, 11439 pollacks@stjohns.edu

### Multivariate normal distribution and testing for means (see MKB Ch 3)

Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 One-sample t-test (univariate).................................................. 3 Two-sample t-test (univariate).................................................

### The Wilcoxon Rank-Sum Test

1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

### Binomial lattice model for stock prices

Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

### Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5

Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.

### 1 Prior Probability and Posterior Probability

Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

### UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Math 431 An Introduction to Probability. Final Exam Solutions

Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont

CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont To most people studying statistics a contingency table is a contingency table. We tend to forget, if we ever knew, that contingency

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test.

HYPOTHESIS TESTING Learning Objectives Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. Know how to perform a hypothesis test

### CHI-SQUARE: TESTING FOR GOODNESS OF FIT

CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity

### Topic 8. Chi Square Tests

BE540W Chi Square Tests Page 1 of 5 Topic 8 Chi Square Tests Topics 1. Introduction to Contingency Tables. Introduction to the Contingency Table Hypothesis Test of No Association.. 3. The Chi Square Test

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### 5 Numerical Differentiation

D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives

### Dongfeng Li. Autumn 2010

Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis

### STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

### Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

### Inference for two Population Means

Inference for two Population Means Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison October 27 November 1, 2011 Two Population Means 1 / 65 Case Study Case Study Example

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Objective. Materials. TI-73 Calculator

0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI-73 Calculator Integer Subtraction What s the Difference? Teacher