Statistical InferenceStatistical Inference involves estimating a population parameter (mean) from a sample that is taken from the population. Inference may also be concerned with the difference between populations on a given parameter (mean depression scores, etc).
Issues In InferenceThe task of inference is to draw conclusions about a parameter (characteristic of the population) from a sample statistic.Because of sampling variation, we can never know if our inferences are exactly correct.The key to any problem in statistical inference is to discover what sample variables will occur in repeated sampling, and with what probability.
Hypothesis TestingA hypothesis is a testable statement about a population parameter. A hypothesis is tested, and based on the outcome, is retained or rejected (never say proven!).Null Hypothesis: any observed difference between our expected and observed values (or between groups) is due to chance.Alternative Hypothesis: an observed difference (between expected and observed values, or between groups) is too drastic to be able to be explained by chance.
Hypothesis TestingWe want to test out a new therapy method on people with depression. Null Hypothesis: The observed difference between the depression in this group and national depression rates can be explained by chance variation in the data.Alternative Hypothesis: The observed difference is too drastic to be able to be explained by chance.We test the null hypothesis. We ask what type of results are likely, if the sample group is no different than the population group. If our sample outcome is unlikely, assuming it should look like the population outcome, than we reject the null.
ExampleLets say that we are interested in whether someone has been using a weighted coin while gambling. Our null hypothesis will be that this coin should not have outcomes that differ from what we expect of a fair coin.This coin comes up 5 heads in a row. It is possible that someone will get 5 heads in 5 tosses, but how likely is it? There is only about a .05 chance that someone will get 5 heads in a row. We reject the null hypothesis.
Testing a Hypothesis about a Single MeanRemember this example from last week?Lets assume we have collected data from a group of 100 1st grade kids in schools across the country who have been listening to classical music during the school day. We know that the national average for IQ for 1st graders is a mean (X) of 100, SD (X) of 15. Does classical music influence IQ?
Testing a Hypothesis about a Single MeanWe are interested in testing the Null Hypothesis, that there is no difference between the sample and the population that cannot be explained by chance. H0 : X = 100The Alternative Hypothesis proposes that there is a difference between the sample and population that cannot be explained by chance. H1 : X 100note that 100 indicates that this score could either be higher or lower than the average.
Testing a Hypothesis about a Single Mean: alpha levelsHow do we decide whether we retain or reject the Null Hypothesis?If we get a mean sample value that falls into a category we dont expect to see often by chance, we can reject the null. Common practice is to reject the null if the sample mean is so deviant that its probability of occurrence is .05 (or .01) or less.This criterion is called the alpha level (). We would note: = .05
Testing a Hypothesis about a Single Mean: rejection regionsIn our case, we want to find out what z-score values will separate 95% of the general population distribution from the remaining 5% at both tails of the distribution. Since we are testing the null hypothesis at = .05, we will only reject H0 if the obtained sample mean is so deviant that it falls in the upper or lower 2.5% of the distribution (the extreme 5% of cases).The portion of the distribution that includes sample mean values that lead to rejection of the null hypothesis are called regions of rejection. The z-scores that separate these areas are called the critical values (zcrit).
Testing a Hypothesis about a Single Mean: resultsIn our case, our zcrit values are 1.96 and + 1.96 (table D.2).Next we compare our test statistic to these critical values.We collect data from 100 students who listened to classical music and have a mean IQ = 105.
This means that our sample average is 3.33 SEs above our expected value. This observed z-score (zobs = 3.33) falls into our region of rejection. What does this mean? What is our conclusion? Have we proven that the mean is greater than 100 in these kids?
The Five Steps of Hypothesis Testing!Formulate the null (H0) and alternative (H1) hypotheses.Identify the relevant test statistic (z score of the sample mean, or z-test) that will be used to discriminate between different hypotheses about the population parameter of interest (mean). Identify the sampling distribution of the statistic under consideration (for z-test - sampling distribution of the mean, and standard error).Determine the level and critical rejection region in which test statistics that warrant rejection of H0 will fall such as p < .05).Conduct the experiment and collect data, then report the observed test statistic (zobs). Use this test statistic in order to make a statistical decision about H0, using the decision rule: if the test statistic lies in the critical rejection region (or if p < .05), then reject H0 (support for H1). If not, retain H0 (lack of support for H1). Make a theoretical conclusion.
Example using the steps of hypothesis testingSuppose you are a clinical psychologist and that you have collected data from a group of 50 people who have gone through your patented program to reduce Ophidiophobia (snake phobia). You have data from a national study of snake phobics that indicate that their pulse rates per second in the presence of a snake are normally distributed with .The group of people that went through the program had their pulse rates measured in the presence of a snake. Result: Does this program influence Ophidiophobia?
Example using the steps of hypothesis testingStep 1 -H0 : H1 : Step 2 - test statistic to be used?Step 3 - sampling distribution to be used ?Step 4 - a = ? one-tailed or two-tailed test? zcrit values = ?Step 5 -calculate (zobs) statistical decision? theoretical conclusion?
But waitWhat if we had approached this analysis as a one-tailed test? Would our results have been different if we only looked at whether the program reduced Ophidiophobia?Then we would have had an a = .05 at one tail, with a zcrit of 1.64 (see table D.2).
What would have been our statistical decision and theoretical conclusion?zobs = -1.72
The Null HypothesisThe favored null hypothesis is held innocent unless proved guilty, while the alternative hypothesis is held guilty until no other choice remains but to judge it innocent.W. W. Rozeboom, 1960
Assumptions for inference using a single mean (z-test)A random sample has been drawn from the population.The sampling has been drawn with replacement.The sampling distribution of X follows the normal curve (central limit theorem).The standard deviation of scores in the population is known.Problem do we always know the population standard deviation? Not usually! (this is dealt with in PSY 321!)
Types of ErrorsWhen we are making statistical decisions like these, there are two types of error that can result:
True State of Null HypothesisStatistical decisionH0 TrueH0 FalseReject H0Type I Error (Rejecting when we shouldnt)Correct RejectionDo not Reject Reject H0Correct Failure to RejectType II Error (Failing to Reject when we should).
Types of ErrorsThe probability of a Type I error is designated by alpha (a) and is called the Type I error rate.Remember that a is also called the significance level, and is set by the experimenter this significance level is chosen in such a way to reduce the probability of a Type I error.The probability of a Type II error (the Type II error rate) is designated by (). A Type II error is only an error in the sense that an opportunity to reject the null hypothesis correctly was lost. It is not an error in the sense that an incorrect conclusion was drawn since no conclusion is drawn when the null hypothesis is not rejected.