class: center, middle, inverse, title-slide # Confidence Interval for Difference of Two Means ### Dr. Dogucu --- layout: true <div class="my-header"></div> <div class="my-footer"> Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div> --- ## Confidence Interval Review Confidence Interval = point estimate `\(\pm\)` critical value `\(\times\)` standard error of the estimate | | point estimate | critical value | standard error of the estimate | |-------------------------------|-----------------------|:--------------:|---------------------------------------------------| | single proportion | `\(p\)` | z* | `\(\sqrt{\frac{p(1-p)}{n}}\)` | | difference of two proportions | `\(p_1-p_2\)` | z* | `\(\sqrt{\frac{p_1(1-p_1)}{n}+\frac{p_2(1-p_2)}{n}}\)` | | single mean | `\(\bar x\)` | `\(t*_{df}\)` | `\(\sqrt{\frac{s^2}{n}}\)` | | difference of two means | `\(\bar x_1 - \bar x_2\)` | `\(t*_{df}\)` | `\(\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)` | --- class: middle ## Conditions 1. Independence: Within each group data have to be independent from each other. The two groups have to be independent from one another. 2. Normality: We check for normality for each group. --- class: center middle ## Confidence Interval for Difference of Two Means `\(\bar x_1 - \bar x_2 \pm t^*_{df} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)` --- class: middle ## Example Onnasch, L., & Roesler, E. (2019). Anthropomorphizing Robots: **The Effect of Framing in Human-Robot Collaboration.** Proceedings of the Human Factors and Ergonomics Society Annual Meeting, 63(1), 1311–1315. https://doi.org/10.1177/1071181319631209 -- **Experiment**: A humanoid robot supports the participant of the study to solve a math puzzle. Do participants perceive the robots differently if the robots are described with anthropomorphic framing or functional framing? --- Response variable: Perception of the robot (humanness, eeriness, acceptance) .pull-left[ **Anthropomorphic framing** - the robot has a name - has a personal story - has a favorite color and hobbies - pronoun: him ] .pull-right[ **Functional framing** - height, weight - pronoun: it ] --- Response variable: Perception of the robot (humanness, eeriness, acceptance) .pull-left[ **Anthropomorphic framing** - the robot has a name - has a personal story - has a favorite color and hobbies - pronoun: him `\(n_1=20\)` `\(\bar x_1 = 3.18\)` `\(s_1 = 0.57\)` ] .pull-right[ **Functional framing** - height, weight - pronoun: it <br> `\(n_2=20\)` `\(\bar x_2 = 3.07\)` `\(s_2 = 0.29\)` ] --- class: middle ## Conditions - Independence within groups - Independence between groups - Normality --- class: center middle ## CI `\(\bar x_1 - \bar x_2 \pm t^*_{df} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)` --- ## 95% CI .pull-left[ ```r xbar1 <- 3.18 xbar2 <- 3.07 point_estimate <- xbar1-xbar2 point_estimate ``` ``` ## [1] 0.11 ``` ```r s1 <- 0.57 s2 <- 0.29 n1 <- 20 n2 <- 20 ``` ] .pull-right[ ```r critical_value <- qt(0.975, df = 19) # we use the smaller group's n-1 critical_value ``` ``` ## [1] 2.093024 ``` ```r se <- sqrt((s1^2)/n1 + (s2^2)/n2) se ``` ``` ## [1] 0.1430035 ``` ] --- ## 95% CI ```r point_estimate - critical_value*se ``` ``` ## [1] -0.1893098 ``` ```r point_estimate + critical_value*se ``` ``` ## [1] 0.4093098 ``` 95% CI = (-0.1893098 , 0.4093098)