Add HW2 and HW3

HW2.Rmd

@@ -9,8 +9,90 @@ output:
     df_print: paged
 ---
 
+```{r}
+library(tidyverse)
+dd <- beaver2
+```
+
 # Question 1
 
+mu_0 = mean temperature when activ = 0
+
+mu_1 = mean temperature when activ = 1
+
+$$ H_0 : \mu_0 = \mu_1, \space H_1 : \mu_0 \neq \mu_1 $$
+
+```{r}
+t.test(temp ~ activ, data = dd)
+```
+
+Reject H_0, accept H_1, we conclude that mean temperatures differ by activity level
+
+Now manually:
+
+```{r}
+summary_stats <- dd %>%
+  group_by(activ) %>%
+  summarise(
+    n = n(),
+    mean_temp = mean(temp),
+    var_temp = var(temp)
+  )
+
+summary_stats
+```
+
+Compute Standard Error and t stat
+
+```{r}
+x0 <- summary_stats$mean_temp[summary_stats$activ == 0]
+x1 <- summary_stats$mean_temp[summary_stats$activ == 1]
+
+s0 <- summary_stats$var_temp[summary_stats$activ == 0]
+s1 <- summary_stats$var_temp[summary_stats$activ == 1]
+
+n0 <- summary_stats$n[summary_stats$activ == 0]
+n1 <- summary_stats$n[summary_stats$activ == 1]
+
+SE <- sqrt(s0 / n0 + s1 / n1)
+
+t_stat <- (x0 - x1) / SE
+t_stat
+```
+
+Compute DF
+
+```{r}
+df <- (s0/n0 + s1/n1)^2 /
+  ((s0/n0)^2/(n0-1) + (s1/n1)^2/(n1-1))
+
+df
+```
+
+Compute p-value
+
+```{r}
+p_value <- 2 * pt(-abs(t_stat), df)
+p_value
+```
+
+This p value also matches the conclusion that t.test reaches, reject H_0, accept H_1. We conclude that mean temperatures differ by activity level
 
 # Question 2
 
+```{r}
+dd <- iris %>%
+  filter(Species %in% c("setosa", "versicolor"))
+```
+
+mu_0 = mean Sepal.Length for setosa
+
+mu_1 = mean Sepal.Length for versicolor
+
+$$ H_0 : \mu_0 = \mu_1, \space H_1 : \mu_0 \neq \mu_1 $$
+
+```{r}
+t.test(Sepal.Length ~ Species, data = dd)
+```
+
+p-value \< 0.05, reject H_0, accept H_1. This indicates a statistically significant difference in mean Sepal.Length between setosa and versicolor.