Two sample t-test

Two sample t-test is used to compare the mean of two groups. Here, you are testing:

where data1 and data2 are vectors that include data from different categories. Alternatively, you can run the following code:

where numerical is the column name with the data and categorical is the column name that includes the group of the observations.

Darwin’s theory of natural selection

In the search for evidence supporting Charles Darwin’s theory of natural selection, biologists Peter and Rosemary Grant and colleagues in the 1980s caught and measured all the birds from more than 20 generations of finches on the Galapagos island of Daphne Major. In one of those years, 1977, a severe drought caused vegetation to wither, and the only remaining food source was a large, tough seed which the finches ordinarily ignored. Were the birds with larger and stronger beaks for opening these tough seeds more likely to survive that year and did they tend to pass this characteristic to their offspring?

The grants measured beak depths (height of the beak at its base) of all 751 Daphne Major finches the year before the drought (1976) and all 89 finches captured the year after the drought (1978). Is there evidence of a difference between the population distributions of beak depths in 1976 and 1978?

df <- read.csv('Data/Darwin.csv')
head(df)

We will be plotting this data at first:

boxplot(Depth~Year, data=df, main='Finch Beacks', xlab='Beak Depths', col='orange',
        ylab='Year',border = 'orange3', horizontal=TRUE)

We will be conducting a two sample t-test:

t.test(Depth~Year, data=df, equal.var=FALSE)

## 
##  Welch Two Sample t-test
## 
## data:  Depth by Year
## t = -5.2049, df = 117, p-value = 8.394e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.7418844 -0.3329222
## sample estimates:
## mean in group 1976 mean in group 1978 
##           9.600799          10.138202

This output shows the sample mean of two groups, p-value from the two sample t-test and the 95% confidence interval of the difference in means.

Conclusion:

These data provide evidence that the mean beak depth increased from 1976 to 1978. The 1978 (post-drought) mean was estimated to exceed the 1976 (pre-drought) mean by 0.54mm (95% CI = [0.34, 0.74]).