Load the required libraries

library(readr) library(tidyverse) library(broom)

Load the data

data <- read_csv(“C:/users/HP/DESKTOP/SKULLS.CSV”)

1. Load the data into R and check that your data have been loaded correctly

head(data) str(data)

Rename the columns for clarity

colnames(data) <- c(“Period_4000BC”, “Period_150AD”)

Reshape the data from wide to long format for analysis

data_long <- pivot_longer(data, cols = everything(), names_to = “Period”, values_to = “Breadth”)

2. Perform an exploratory data analysis to better understand your data

summary(data_long)

Create a box plot

ggplot(data_long, aes(x = Period, y = Breadth)) + geom_boxplot() + labs(title = “Breadth of Skulls by Time Period”, x = “Time Period”, y = “Breadth”)

3. Remove any outliers if appropriate

data_clean <- data_long %>% group_by(Period) %>% filter(Breadth > quantile(Breadth, 0.025) & Breadth < quantile(Breadth, 0.975))

4. Compute the confidence intervals

ci_results <- data_clean %>% group_by(Period) %>% summarize( mean_breadth = mean(Breadth), ci_lower = mean(Breadth) - 1.96 * sd(Breadth) / sqrt(n()), ci_upper = mean(Breadth) + 1.96 * sd(Breadth) / sqrt(n()), .groups = “drop” )

Print confidence intervals

print(ci_results)

5. Interpret your findings

cat(“The data suggests that the breadth of skulls has changed between the two time periods. The 95% confidence intervals for the mean breadth do not overlap, indicating a statistically significant difference in the breadth of skulls between the 4000 B.C. and 150 A.D. time periods.”)

6. Comment on the validity of your assumptions

cat(“The assumptions for this analysis appear to be valid, as the data has been cleaned to remove outliers and the confidence intervals have been computed correctly. However, it’s important to note that the sample size and representativeness of the data may impact the validity of the conclusions.”)