Psyc 235: Introduction to Statistics DON’T FORGET TO SIGN IN FOR CREDIT!

Stuff Thursday: office hours  hands-on help with specific problems Next week labs:  demonstrations of solving various types of hypothesis testing problems

  Population Sample Sampling Distribution size = n (of the mean)

Descriptive vs Inferential Descriptive  describe the data you’ve got  if those data are all you’re interested in, you’re done. Inferential  make inferences about population(s) of values  (when you don’t/can’t have complete data)

Inferential Point Estimate Confidence Interval Hypothesis Testing  1 population parameter  z, t tests  2 pop. parameters  z, t tests on differences  3 or more?...  ANOVA!

Hypothesis Testing 1.Choose pop. parameter of interest (ex:  ) 2.Formulate null & alternative hypotheses assume the null hyp. is true 3.Select test statistic (e.g., z, t) & form of sampling distribution based on what’s known about the pop., & sample size

Defining our hypothesis H 0 = the Null hypothesis  Usually designed to be the situation of no difference  The hypothesis we test. H 1 = the alternative hypothesis  Usually the research related hypothesis

Null Hypothesis (~Status Quo) Examples: Average entering age is 28 (until shown different) New product no different from old one (until shown better) Experimental group is no different from control group (until shown different) The accused is innocent (until shown guilty)

- H a is the hypothesis you are gathering evidence in support of. - H 0 is the fallback option = the hypothesis you would like to reject. - Reject H 0 only when there is lots of evidence against it. - A technicality: always include “=” in H 0 - H 0 (with = sign) is assumed in all mathematical calculations!!!

Decision Tree for Hypothesis Testing Population Standard Deviation known? Yes No Pop. Distribution normal? n large? (CLT) Yes No Yes No Yes No Yes No z-score Can’t do it t-score Test stat. Standard normal distribution t distribution

Selecting a distribution

Hypothesis Testing 1.Choose pop. parameter of interest (ex:  ) 2.Formulate null & alternative hypotheses assume the null hyp. is true 3.Select test statistic (e.g., z, t) & form of sampling distribution based on what’s known about the pop., & sample size

Hypothesis Testing 4.Calculate test stat.: 5.Note: The null hypothesis implies a certain sampling distribution 6.if test stat. is really unlikely under Ho, then reject Ho HOW unlikely does it need to be? determined by 

Three equivalent methods of hypothesis testing (  =significance level) p-value: prob of getting test stat at least as extreme if Ho really true.

Hypothesis Testing as a Decision Problem Great! Type II Error Great! Type I Error Power: 1 – P(Type II error) Our ability to reject the null hypothesis when it is indeed false Depends on sample size and how much the null and alternative hypotheses differ

ERRORS Type I errors (  ): rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit. Type II errors (  ): failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit.

Type I and Type II errors Power (1-   

ANOVA: Analysis of Variance a method of comparing 3 or more group means simultaneously to test whether the means of the corresponding populations are equal  (why not just do a bunch of 2-sample t- tests?...)  inflation of Type I error rate

ANOVA: 1-Way You have sample data from several different groups “One-way” refers to one factor. Factor = a categorical variable that distinguishes the groups. Level (group) of the factor refers to the different values that the categorical variable can take.

ANOVA: 1-Way Examples of Factors & groups:  Factor: Political Affiliation  groups: Democrat, Republican, Independent  X=annual income  Factor: Studying Method  groups: Re-read notes, practice test, do nothing (control)  X=score on exam

ANOVA: 1-Way So you’ve got 3(+) sets of sample data, from 3 different populations. You want to test whether those 3 populations all have the same mean (  ) Null Hypothesis:  H 0 :  1 =  2 =  3 (all pop. means are same)  H 1 : all pop. means are NOT the same! [draw examples on chalkboard]

ANOVA: Assumptions Normality  populations are normally distributed Homogeneity of variance  populations have same variance (  2 ) 1-Way “Independent Samples”:  groups are independent of each other

ANOVA: the idea Two ways to estimate  2  MSB: Mean Square Between Group (aka MSE: MS Error)  based on how spread out the sample means are from each other.  Variation Between Samples  MSW: Mean Square Within Group  based on the spread of data within each group  Variation within Samples If the 3(+) populations really do have same mean, then these 2 #s should be ~ the same If NOT, then MSB should be bigger.

ANOVA: calculating MSB: Variation between samples  (sample size) * (variance of sample means)  if sample sizes are the same in all groups  note: use the “sample variance” formula MSW: Variation within samples  (mean of sample variances)

ANOVA: the F statistic So how to compare MSB and MSW? Under H 0 : F≈1 So calculate your F test statistic and compare to F distribution, see if it falls in region of rejection. [chalkboard] note: F one-tailed!

ANOVA: F & df F distribution requires specification of 2 degrees of freedom values DFn: degrees of freedom numerator:  (# of groups) - 1 DFd: degrees of freedom denominator:  (total sample size (N)) - (# of groups)

ANOVA: example Groups: adults w/ 3 different activity levels X=% REM sleep MSB=(sample size)(variance of sample means)=... MSW=(mean of sample variance)=... F=MSB/MSW=... dfn=# groups - 1=... dfd=Ntotal-#groups=... Fcritical=...p-value=...