class: center, middle, inverse, title-slide # CLT based inference ### Dr. Çetinkaya-Rundel ### 2018-04-02 --- ## Announcements - Midterm assigned at noon Monday, due Friday at noon - Mostly modeling + inference - But also some exploratory data analysis - OH tomorrow: 10:30am - noon - Team meetings - What do I owe you? - One other extra credit assignment - Some regrades - HW5 feedback or key --- class: center, middle # Inference methods based on CLT --- ## What is the CLT? The Central Limit Theorem tells us the distribution of certain sample statistics if necessary conditions are met. - The distribution of the sample statistic is nearly normal - The distribution is centered at the (often unknown) population parameter - The variability of the distribution is inversely proportional to the square root of the sample size --- ## Inference methods based on CLT If necessary conditions are met, we can also use inference methods based on the CLT: -- - use the CLT to calculate the SE of the sample statistic of interest (sample mean, sample proportion, difference between sample means, etc.) -- - calculate the **test statistic**, number of standard errors away from the null value the observed sample statistic is - Z for proportions - T for means, along with appropriate degrees of freedom -- - use the test statistic to calculate the **p-value**, the probability of an observed or more extreme outcome given that the null hypothesis is true --- ## Z distribution .small[ Also called the **standard normal distribution**: `\(Z \sim N(mean = 0, \sigma = 1)\)` ] .small[ Finding probabilities under the normal curve: ```r pnorm(-1.96) ``` ``` ## [1] 0.0249979 ``` ```r pnorm(1.96, lower.tail = FALSE) ``` ``` ## [1] 0.0249979 ``` ] -- .small[ Finding cutoff values under the normal curve: ```r qnorm(0.025) ``` ``` ## [1] -1.959964 ``` ```r qnorm(0.975) ``` ``` ## [1] 1.959964 ``` ] --- ## T distribution - Also unimodal and symmetric, and centered at 0 - Thicker tails than the normal distribution (to make up for additional variability introduced by using `\(s\)` instead of `\(\sigma\)` in calculation of the SE) - Parameter: **degrees of freedom** - df for single mean: `\(df = n - 1\)` - df for comparing two means: `$$df \approx \frac{(s_1^2/n_1+s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1)+(s_2^2/n_2)^2/(n_2-1)} \approx min(n_1 - 1, n_2 - 1)$$` --- ## T vs Z distributions <!-- --> --- ## T distribution .small[ Finding probabilities under the t curve: ```r pt(-1.96, df = 9) ``` ``` ## [1] 0.0408222 ``` ```r pt(1.96, df = 9, lower.tail = FALSE) ``` ``` ## [1] 0.0408222 ``` ] -- <br/> .small[ Finding cutoff values under the t curve: ```r qt(0.025, df = 9) ``` ``` ## [1] -2.262157 ``` ```r qt(0.975, df = 9) ``` ``` ## [1] 2.262157 ``` ] --- class: center, middle # Example --- ## Relaxing after work .question[ The GSS asks "After an average work day, about how many hours do you have to relax or pursue activities that you enjoy?". Do these data provide convincing evidence that Americans, on average, spend more than 3 hours per day relaxing? Note that the variable of interest in the dataset is `hrsrelax`. ] ```r gss = read_csv("data/gss2010.csv") gss %>% filter(!is.na(hrsrelax)) %>% summarise(x_bar = mean(hrsrelax), med = median(hrsrelax), sd = sd(hrsrelax), n = n()) ``` ``` ## # A tibble: 1 x 4 ## x_bar med sd n ## <dbl> <dbl> <dbl> <int> ## 1 3.68 3. 2.63 1154 ``` --- ## Exploratory Data Analysis ```r ggplot(data = gss, aes(x = hrsrelax)) + geom_histogram(binwidth = 1) ``` ``` ## Warning: Removed 890 rows containing non-finite values (stat_bin). ``` <!-- --> --- ## Hypotheses .question[ What are the hypotheses for evaluation Americans, on average, spend more than 3 hours per day relaxing? ] -- `$$H_0: \mu = 3$$` `$$H_A: \mu > 3$$` --- ## Conditions .question[ What conditions must be satisfied to conduct this hypothesis test using methods based on the CLT? Are these conditions satisfied? ] --- ## Calculating the test statistic Summary stats from the sample: ``` ## # A tibble: 1 x 3 ## xbar s n ## <dbl> <dbl> <int> ## 1 3.68 2.63 1154 ``` And the CLT says: `$$\bar{x} \sim N\left(mean = \mu, SE = \frac{\sigma}{\sqrt{n}}\right)$$` -- .question[ How many standard errors away from the population mean is the observed sample mean? ] -- .question[ How likely are we to observe a sample mean that is at least as extreme as the observed sample mean, if in fact the null hypothesis is true? ] --- ## Calculations ```r (se <- hrsrelax_summ$s / sqrt(hrsrelax_summ$n)) ``` ``` ## [1] 0.07740938 ``` ```r (t <- (hrsrelax_summ$xbar - 3) / se) ``` ``` ## [1] 8.7876 ``` ```r (df <- hrsrelax_summ$n - 1) ``` ``` ## [1] 1153 ``` ```r pt(t, df, lower.tail = FALSE) ``` ``` ## [1] 2.720895e-18 ``` --- ## Conclusion - Since the p-value is small, we reject `\(H_0\)`. - The data provide convincing evidence that Americans, on average, spend more than 3 hours per day relaxing after work. -- .question[ Would you expect a 90% confidence interval for the average number of hours Americans spend relaxing after work to include 3 hours? ] --- ## Confidence interval for a mean `$$point~estimate \pm critical~value \times SE$$` ```r t_star <- qt(0.95, df) pt_est <- hrsrelax_summ$xbar round(pt_est + c(-1,1) * t_star * se, 2) ``` ``` ## [1] 3.55 3.81 ``` .question[ Interpret this interval in context of the data. ] --- ## Built-in functionality in R - There are built in functions for doing some of these tests in R: - However a learning goal is this course is not to go through an exhaustive list of all CLT based tests and how to implement them - Instead you should try to understand how these methods are / are not like the simulation based methods we learned about earlier .question[ What is similar, and what is different, between CLT based test of means vs. simulation based test? ] --- .small[ ```r # HT t.test(gss$hrsrelax, mu = 3, alternative = "greater") ``` ``` ## ## One Sample t-test ## ## data: gss$hrsrelax ## t = 8.7876, df = 1153, p-value < 2.2e-16 ## alternative hypothesis: true mean is greater than 3 ## 95 percent confidence interval: ## 3.552813 Inf ## sample estimates: ## mean of x ## 3.680243 ``` ```r # CI t.test(gss$hrsrelax, conf.level = 0.90)$conf.int ``` ``` ## [1] 3.552813 3.807672 ## attr(,"conf.level") ## [1] 0.9 ``` ]