Assignment 2

STAT3373

library(tidyverse)

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dd <- beaver2

Question 1

mu_0 = mean temperature when activ = 0

mu_1 = mean temperature when activ = 1

\[ H_0 : \mu_0 = \mu_1, \space H_1 : \mu_0 \neq \mu_1 \]

t.test(temp ~ activ, data = dd)

## 
##  Welch Two Sample t-test
## 
## data:  temp by activ
## t = -18.548, df = 80.852, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
## 95 percent confidence interval:
##  -0.8927106 -0.7197342
## sample estimates:
## mean in group 0 mean in group 1 
##        37.09684        37.90306

Reject H_0, accept H_1, we conclude that mean temperatures differ by activity level

Now manually:

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

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

## [1] -18.54789

Compute DF

df <- (s0/n0 + s1/n1)^2 /
  ((s0/n0)^2/(n0-1) + (s1/n1)^2/(n1-1))

df

## [1] 80.85226

Compute p-value

p_value <- 2 * pt(-abs(t_stat), df)
p_value

## [1] 7.269112e-31

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

dd <- iris %>%
  filter(Species %in% c("setosa", "versicolor"))

mu_0 = mean Sepal.Length for setosa

mu_1 = mean Sepal.Length for versicolor

\[ H_0 : \mu_0 = \mu_1, \space H_1 : \mu_0 \neq \mu_1 \]

t.test(Sepal.Length ~ Species, data = dd)

## 
##  Welch Two Sample t-test
## 
## data:  Sepal.Length by Species
## t = -10.521, df = 86.538, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group setosa and group versicolor is not equal to 0
## 95 percent confidence interval:
##  -1.1057074 -0.7542926
## sample estimates:
##     mean in group setosa mean in group versicolor 
##                    5.006                    5.936

p-value < 0.05, reject H_0, accept H_1. This indicates a statistically significant difference in mean Sepal.Length between setosa and versicolor.