Stat 217 – Day 24 Analysis of Variance Have yesterday’s handout handy

Last Time – Chi-square Tests When?  Two categorical variables (comparing several population proportions) Research question  Null hypothesis:  1 =  2 = … =  I (no association between the two variables)  Alternative hypothesis: at least one  differs (there is an association between the two variables)

How?  Compare observed counts to expected counts (under the null hypothesis)  Compare test statistic to “chi-square” distribution, p-value  Output: observed counts, expected counts, cell contributions, test statistic, p-value Technical conditions:  Randomness: random sampling or random assignment  Sample size: all expected counts > 5

Activity 25-3 (e)-(g) Can apply to randomized experiment as well Ho: The population proportions of potential customers who would leave a tip (or the probability is the same regardless of the type of card they receive) No association between type of card and whether or not tip Ha: not the same (is an association) Conclusion: Is significant evidence (.007 <.05) that the type of card affects the likelihood of someone receiving a tip, at least for this waitress, this coffee bar

Follow-up Analysis Observed more tips with the joke card than expected and fewer tips with the ad than expected Left tip Didn’t leave tip

Activity 25-2 (p. 511) What if have a non-binary response variable? Same thing! (a) Ho: the population distributions of happiness were the same all three years no association between happiness level and year Ha: the population distributions were not the same (is an association) (b) X 2 = (df = 4), p-value =.000 (c) Strong evidence of a change in at least one of these population distributions

Activity 25-2 Where are the differences (descriptively)? Fewer “not too happy” in 1998 than expected. More “not too happy” in 1972 than expected.

Activity 25-4 (b) Data collection: one sample, both variables recorded simultaneously (not independent random samples or randomized experiment) Ho: no association between happiness level and political inclination in population Ha: is an association Same analysis!

Activity 25-4 Conclusion We have strong evidence (p-value =.001 <.05) that there is a relationship between political inclination and happiness level in the population of adult Americans.

Example: Disability Employment Researchers conducted a study in the 1980s that examined whether physical handicaps affect people’s perceptions of employment qualifications. They randomly assigned 70 undergraduates to watch a videotaped job interview and to rate the candidate’s qualifications on a 1-10 scale. The videos used the same actors and script each time but the job applicant appeared with different handicaps: no disability, leg amputation, crutches, hearing impairment, and wheelchair confinement. TIA

Example (a) 70 undergraduate students EV: type of disability RV: job rating one quantitative and one categorical

Example Let  1 represent the average employment rating by the “population” of students watching the “no disability” video. Similarly for  2,  3,  4, and  5 Ho:  1 =  2 =  3 =  4 =  5 (no disability effect) no relationship between type of disability and employment rating Ha: not all means equal (a disability effect)

Discussion of descriptive statistics For these 70 students, appeared to be some differences in how rated individuals with different disabilities. The average rating for crutches was highest (5.92), hearing the lowest (4.05), but also a fair bit of overlap between the distributions. Distributions are reasonably symmetric with similar spread without any visual outliers.

Example (with 3 groups…) Not significant Significant

Test Statistic Need a way to compare the variability between the groups while taking into account the expected “natural variation” Do the sample means differ more than we would expect “by chance”? “F statistic”  Numerator – compares 5 means to overall mean  Denominator – an “average” standard deviation  More details in “topic 33” online

Minitab One-way ANOVA: SCORE versus DISABILITY Source DF SS MS F P DISABILITY Error Total Small p-value, so we reject the null hypothesis and conclude that there is a difference, on average, in employment ratings given across disability types  Cause and effect…  Applies to population of undergraduate students? Variation between disability groups Variability within disability groups, “s 2” s=5.23… ratio

Technical conditions 1) Independent SRSs from each population or randomized comparative experiment 2) Each population follows a normal distribution 3) Each population has the same standard deviation 1.794/1.482 < 2

To Turn in with one partner  Your answer to (g) of Example 1 as well as Example 2 For Wednesday  Read about technical conditions, answer (h) and (i) on handout, read about ANOVA/chi-square parallels