Comparing two means

# Load the mosaic package
library(tidyverse)

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Scenario: Arachnophobia

24 arachnophobe volunteers are randomly assigned to spend 10 minutes one of two rooms:

–>

     Click the tabs to see what’s in each room…

Room A

The room contains a real spider (in a cage).

Room B

The room contains only a photo of that same spider.

Does the real spider elicit more anxiety than the photo?

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Data Exploration

     Click code to see how the data were entered –>

# Enter the data manually
spider <- tibble(
  group = c( rep("photo", 12), rep("real", 12) ),
  anxiety = c(30, 35, 45, 40, 50, 35, 55, 25, 30, 45, 40, 50, 
              40, 35, 50, 55, 65, 55, 50, 35, 30, 50, 60, 39))

# Display some data (ordered by anxiety levels)
spider %>%
  arrange(anxiety)

Summary statistics

# Display summary statistics

spider %>%
  group_by(group) %>%
  summarize(mean = mean(anxiety),
            median = median(anxiety),
            sd = sd(anxiety),
            n = n())

Dotplot

spider %>%
  ggplot(aes(x = group, y = anxiety)) +
  geom_dotplot(binaxis = "y", binpositions="all", stackdir = "center", fill = "steelblue", color = "white") +
  scale_y_continuous(breaks=seq(20, 70, 10), minor_breaks=NULL) +
  theme(axis.title.x = element_text(size = 12, color = "#777777")) +
  theme(axis.text.x = element_text(size = 12)) +
  theme(axis.title.y = element_text(size = 12, color="#777777")) +
  theme(axis.text.y = element_text(size = 12)) +
  labs(
    title = "Anxiety by group"
  )

Boxplot

spider %>%
  ggplot(aes(y = anxiety, x = group)) +
  geom_boxplot(fill = "white", color = "black", alpha = 0.9) +
  geom_dotplot(binaxis = "y", stackdir = "center", fill = "steelblue", color = "white", alpha = 0.6) +
  scale_y_continuous(breaks=seq(20, 70, 10), minor_breaks=NULL) +
  theme(axis.title.x = element_text(size = 12, color = "#777777")) +
  theme(axis.text.x = element_text(size = 12)) +
  theme(axis.title.y = element_text(size = 12, color="#777777")) +
  theme(axis.text.y = element_text(size = 12)) +
  labs(
    title = "Anxiety by group"
  )

.

1. It appears as though the subjects with the real spider experienced greater anxiety. Explain why this does not prove the real spider elicits more anxiety than the photo of the spider.

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Model

2. State the null and alternative hypotheses for this study

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3. blah

blah

4. blah

Let’s construct a model that could have generated the data in this study. First, we’ll define:

• \(y_{ig}=\) the anxiety of subject i in group g

• \(\mu=\) a constant, overall mean anxiety level all humans possess

• \(\alpha _{g}=(\mu _{g}-\mu )=\) the change in anxiety associated with being assigned to group g

• \(\epsilon_{ig}=(y_{ig}-\mu _{g})=\) errors or deviations in anxiety among individuals in a group (due to other factors not analyzed in this study).

With this notation, we can write our full model: \(y_{ig}=\mu +\alpha _{g}+\epsilon_{ig}\).