Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 18 Hypothesis testing of the mean Why I don t use herbal medicines SUPPOSE HYPOTHESIS TESTING we think the mean value of a population of SAT scores is 455 in hypothesis testing we consider how reasonable a hypothesis is given the data that we have we can take a sample of the population and calculate the sample mean of SAT scores X 535 we can make some statement about how rare it is to get X 535 what we did two lectures ago or we can make a statement about how unreasonable it is that our original thought is true if the hypothesis is reasonable consistent with the data we assume it is true if the hypothesis is unreasonable inconsistent with the data we assume it is false deciding on what hypotheses to test is critically important 2 3 HYPOTHESIS TESTING HYPOTHESIS NULL HYPOTHESIS four steps conjecture about one or more population parameters H0 is the assumption of no relationship or no difference e g e g H0 no relationship between variables 1 State the hypothesis 2 Set the criterion for rejecting the hypothesis 3 Compute the test statistic 4 Decide about the hypothesis 455 3 5 r 0 76 in inferential statistics we always test the null hypothesis H0 H0 no difference between treatment groups the alternative hypothesis Ha is the other possibility e g H0 455 Ha 455 does not say what is but says what it is not 4 5 6 NULL HYPOTHESIS NULL HYPOTHESIS STATE THE HYPOTHESIS what s wrong with herbal medicines often times almost always the goal of statistical research is to reject the null hypothesis so that the only alternative is to accept the Ha before doing anything else we need to make certain that we understand the tested hypothesis nothing necessarily but I don t know that they are any good and they may be bad lots of reports that they help people but how can they be sure need to start by assuming that a medicine does nothing and prove that the assumption is false anecdotal reports are just about worthless similar to an indirect proof e g show that the angles of a triangle sum to 180o by assuming that they do not and then finding a contradiction why this approach it is much easier to show that something is false H0 than to show that something is true Ha for the SAT example H0 455 Ha 455 sometimes this is the most difficult step in designing an experiment to start we will worry only about hypotheses about the population mean understanding a relationship between variables or differences between groups often requires many experiments 7 8 9 CRITERION DECISIONS ERRORS we will examine the data to see if we should reject or accept H0 we will do that by comparing the sample mean X to the hypothesized value of the population mean after deciding to reject or not reject H0 there are four possible situations two types of errors the bottom line is whether X is sufficiently different from to reject H0 but we have to consider three things to quantify the term sufficiently different errors in hypothesis testing A true hypothesis is rejected Type I error when we reject a true null hypothesis A true hypothesis is not rejected Type II error when we do not reject a false null hypothesis A false hypothesis is rejected State of nature Decision made H0 true H0 false Reject H0 Type I error Correct decision Do not reject H0 Correct decision Type II error A false hypothesis is not rejected errors are unavoidable we want to minimize the probability of making errors given the particular data set we have level of significance generally decreasing the probability of making one type of error increases the probability of making the other type of error region of rejection 10 11 12 USE SIGNIFICANCE LEVEL REGION OF REJECTION suppose you have a new untested and expensive treatment for cancer alpha level is a probability it identifies how much risk of Type I error we are willing to take rejecting H0 when it is true you run a test to judge whether the drug is better than existing drugs indicates probability of Type I error frequently we choose 0 05 or 0 01 if you reject H0 indicating that the drug is more effective when in fact it is not people will spend a lot of money for no reason Type I error that is the corresponding decision to reject H0 may produce a Type I error 5 or 1 of the time if you fail to reject H0 indicating that the drug is not effective when in fact it is people will not use the drug Type II error a statement about how much error we will accept usually chosen before the data is gathered depends upon use of the analysis consider our example of SAT scores H0 455 suppose we have also been told the following 100 and our sample size is n 144 scientific research tends to focus on avoiding Type I errors 13 14 15 REGION OF REJECTION REGION OF REJECTION REGION OF REJECTION we know that the sampling distribution is area under the curve represents the probability of getting the corresponding sample means given that is where specified in H0 we shade in the extreme percentage of the sampling distribution the extreme tails of the sampling distribution correspond to what should be very rare sample means if the H0 is true if our sample mean X is in the region of rejection we reject H0 because it is unlikely that we would get such a sample mean for the corresponding 1 Normal 2 Has a mean of 455 if H0 is true 3 Has a standard error of the mean 100 X 8 33 n 144 0 04 0 04 0 04 0 03 0 03 0 03 0 02 0 02 0 02 0 01 0 01 0 01 0 0 400 0 420 440 460 Sample Mean 400 420 440 460 480 called the region of rejection 480 500 400 420 440 Sample Mean 16 17 460 Sample Mean 500 18 480 500 CRITICAL VALUES CRITICAL VALUES TEST STATISTIC values of sample means at the beginning of the region of rejection critical values represent the beginning of the region of rejection NOTE is split up in each tail for 0 05 we find from the standard normal table that the critical values in z scores are 1 96 if the z score for X is beyond 1 96 z scores from 455 it is very very unlikely to have occurred if the H0 is true we have the following data called a two tailed or non directional test 0 4 0 04 n 144 sample size 0 3 0 03 X observed value for sample statistic 0 2 0 02 0 1 0 01 0 0 4 400 420 440 460 480 455 H0 2 0 2 4 100 value of the standard deviation of the population X 500 Sample Mean X 8 33 standard error calculated earlier from …