Descriptive and Inferential Statistics Descriptive statistics The science of describing distributions of samples or populations Inferential statistics The science of using SAMPLE statistics to make INFERENCES or DECISIONS about population parameters

Inferential Statistics Examples Quantitative (Confidence Intervals) Using sample mean X to estimate population mean μ Using sample stdev S to estimate population stdev σ Qualitative (Hypothesis Testing) Make Conclusions about behavior in population Most widely used inferential statistic

Are these people homeless?

Possible Cause #1 Have homes and lazy owners Comes from distribution of dogs with homes Possible Cause #2 Homeless Comes from distributions of dogs without homes Why are these people so scruffy? Less Scruffy More Scruffy People with homes People without homes These people Distribution of Scores

Why are these people so scruffy? People with homes People without homes Distribution of Scores Less Scruffy More Scruffy People with homes People without homes These dogs Distribution of Means (N=25)

Cause of Differences Null hypothesis (H 0 ) Alternate hypothesis (H 1 ) Goal is to REJECT NULL HYPOTHESIS Show probability of null hypothesis true as low Hypothesis Testing

When do we reject Null Hypothesis? When we get an event which is unlikely? How big is the red area (rejection region)? Hypothesis Testing

When do we reject Null Hypothesis? Rejection Region =.05 (5%) Why? Why Not? What if.50? What if ? Hypothesis Testing

By using different types of distributions! Single sample Z (easy but uncommon) T (harder but more common) F distributions (ANOVA) Chi-square Distributions (Nominal data) How do we calculate probabilities of rejection regions?

Hypothesis Testing: Single sample z test How do we reject the null hypothesis 1) Creating a null hypothesis Example: The Mean IQ of UCLA students is 100 H 0 : μ=100 2) Creating a distribution of outcomes for H 0 : μ=100 Example: A mean distribution (σ X =10,,N=25)

Hypothesis Testing: Single sample z test 3) Determine outcomes that we consider “Unlikely” Example:

4) Evaluate if our experimental result is one of those unlikely outcomes Example: (X = 105)

5) Two possible outcomes Reject H 0 Fail to Reject H 0 X = 105 X = 102

Inferential Statistics: Hypothesis Testing If we reject H 0, we have show evidence that it is LIKELY to be untrue. But we might be wrong. How likely is it that we are wrong? Answer: The size of the rejection region = Alpha (α)

Inferential Statistics: Hypothesis Testing If H 0 is true in reality Probability of an error α Probability of no error is α But what if H 0 is not true?

Null hypothesis can wrong in many ways

Inferential Statistics: Hypothesis Testing If null hypothesis is not true, then Error if Fail to reject =Beta (β) Correct if reject = 1- β Much harder to calculate than α because depends on how distributions overlap which is affected by Effect Size Sample Size (N) Standard Deviation (σ X ) Alpha (α) Tails of Test

Hypothesis Testing

Hypothesis can be two tail one tail (right) one-tail (left) (non-directional) (directional) (directional) No prior evidence Prior (μ>a) Prior (μ<a) H 0 : μ=a H 1 : μ NOT =a H 0 : μ<=a H 1 : μ>a H 0 : μ>=a H 1 : μ<a

Hypothesis Testing Errors Type 1 errors: What happens Other researchers use this false result as information Fail to replicate Waste time Type II errors: what happens No research report is created Not much loss to the scientific community Loss to researcher Other researchers can still find the effect

Hypothesis Testing Outline of hypothesis evaluation I. State null and alternative hypotheses II. Set the criterion for rejecting H 0 III. Collect a sample and compute the observed values of the sample statistic and the test statistic IV. Interpret the results

Hypothesis Testing Example Suppose we are interested in the effect of "Absotox" on cognitive ability. Our plan: select 25 participants administer Absotox, Have participants take the Hurlburt Anagrams Test ("HAT"). The score on the HAT is the number of seconds required to solve a series of anagrams. We have used the HAT in many previous studies, and know that in undrugged subjects, HAT scores are normally distributed with mean 90 seconds and standard deviation 20 seconds. Does Absotox affect cognitive ability?

Variable H 0 : µ= 90 sec H 1 : µ ≠ 90 sec I. State null and alternative hypotheses

Sample Statistic H 0 : µ= 90 sec H 1 : µ ≠ 90 sec I. State null and alternative hypotheses

Test Statistic I. State null and alternative hypotheses

H 0 : µ= 90 sec H 1 : µ ≠ 90 sec

II. Set the criterion for rejecting H 0 A. Level of significance? Level of significance =.05 unless otherwise specified. Therefore z cv = 1.96 B. Directional or nondirectional? Nondirectional because problem asks whether Absotox affects ability C. Two illustrations of criterion 1....on sample statistic 2....on test statistic

III. Collect a sample and compute the observed values of the sample statistic and the test statistic HAT Scores (sec): 95, 106, 75, 118, 99, 127, 79, 20,109, 101, 94, 91, 121, 94, 97, 94, 121, 105, 111, 135, 71, 73, 108, 65, 119 So X = 2428 Then = X / n = 2428/25 = sec

So z obs = ( - μ ) / σ = ( )/4.00 = 1.78 III. Collect a sample & compute the observed values of sample statistic & test statistic X

IV. Interpret the results A. Statistically significant? B. If so, practically significant? 1. Effect size 2. Illustrate 3. Consider C. Describe (plain English) "Because the observed value of the sample mean did not exceed its critical value, or, equivalently, because the observed value of the test statistic did not exceed its critical value, we did not reject the null hypothesis. We cannot conclude that Absotox has any effect on cognitive functioning."