Simple Random Sample

Method:

Let x be a random positive integer such that:
\[1\leq x \leq 59\] This selects the poem. From there, count number of lines.
Let y be a random number such that: \[1 \leq y \leq n\] where n is the number of lines.

From here the syllables on line y are counted.

#sample.int(59, 30, replace = TRUE)

poem_list <- c(26, 34, 53, 16, 34, 57, 12, 35, 46, 9, 43, 17, 23, 39, 3, 39, 49, 11, 56, 
               58, 20, 59, 25, 46, 41, 21, 23, 2, 20, 11)

#sample.int(13, 1)

syllable_sample_one <- c(10,8,8,8,7,7,4,12,6,5,8,10,7,8,5,11,8,11,7,8,4,8,4,4,9,11,7,4,11,3)
syllable_sample_two <- c(2,6,4,7,4,3,4,5,3,4,4,4,4,8,6,6,7,4,2,4,7,6,2,4,2,6,4,10,3,5)

cat("The sample standard deviation for sample one is:", round(sd(syllable_sample_one),2),
    "Syllables \n \n") 

## The sample standard deviation for sample one is: 2.51 Syllables 
## 

cat("The sample standard deviation for sample two is:", round(sd(syllable_sample_two),2), 
    "Syllables \n \n")

## The sample standard deviation for sample two is: 1.92 Syllables 
## 

cat("The summary statistics for sample one are: \n")

## The summary statistics for sample one are:

summary(syllable_sample_one)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.000   5.250   8.000   7.433   8.750  12.000

cat("\nThe summary statistics for sample two are: \n")

## 
## The summary statistics for sample two are:

summary(syllable_sample_two)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.000   4.000   4.000   4.667   6.000  10.000

boxplot(syllable_sample_one, syllable_sample_two,
        horizontal = TRUE, 
        xlab = "Line Length (Syllables)", 
        names = c("Stephen Crane", "Partner Sample"))

syllable_df_one = as.data.frame(syllable_sample_one)
ggplot(data = syllable_df_one, aes(x=syllable_sample_one, y = ..count../sum(..count..))) + 
  geom_histogram(binwidth = 1, colour = 'black') + 
  labs(title = "Relative Frequency Histogram of Stephen Crane Poem Line Lengths",
       subtitle = "Poems Selected by Gerald D. McDonald (1964)") +
  scale_x_continuous("Line Length (Syllables)", breaks = 3:12, labels = 3:12) +
  scale_y_continuous("Rel. Frequency", breaks = seq(0,.4,.05)) +
  theme_grey() 

syllable_df_two = as.data.frame(syllable_sample_two)
ggplot(data = syllable_df_two, aes(x=syllable_sample_two, y = ..count../sum(..count..))) + 
  geom_histogram(binwidth = 1, colour = 'black') + 
  labs(title = "Relative Frequency Histogram of Charles Tomlinson Poem Line Lengths") +
       # subtitle = "Poems Selected by Gerald D. McDonald (1964)") +
  scale_x_continuous("Line Length (Syllables)", breaks = 2:10, labels = 2:10) +
  scale_y_continuous("Rel. Frequency", breaks = seq(0,.4,.05)) +
  theme_grey() 

Data Analysis Conclusions:

Building off of the observations made in the previous section, it can be stated that both distributions are very different from each other. This goes beyond shape as well. It would be one thing if that was the main difference but it can also be seen that the Stephen Crane sample line lengths are, on average, far longer than the line length found in the second sample. This can be observed from the summary statistics and box plots which also show a far greater spread in the Stephen Crane sample. The second sample has a more dense distribution of line lengths that tend to be shorter, with a third quartile below even the median of the Stephen Crane sample.

95% Confidence Interval

lower_bound <- mean(syllable_sample_one) - 
  qt(1-0.05 / 2,29) * 
  sd(syllable_sample_one) / sqrt(30)

upper_bound <- mean(syllable_sample_one) + 
  qt(1-0.05 / 2,29) * 
  sd(syllable_sample_one) / sqrt(30)

cat("The double sided 95% confidence interval for the population mean of Stephen Crane",
      "\npoem line lengths in syllables is: \n[",
    round(lower_bound,2), ",", round(upper_bound,2), "]")

## The double sided 95% confidence interval for the population mean of Stephen Crane 
## poem line lengths in syllables is: 
## [ 6.49 , 8.37 ]

In other words, it is known with 95% confidence that the true mean of Stephen Crane poem line lengths is between 6.49 and 8.37 syllables long based on our given sample.

Hypothesis Test

\[H_0: \mu_0 = \mu_1\] \[H_A: \mu_0 \neq \mu_1\] \[\alpha = .05\]

To put this into words, the null hypothesis is that both poets true average line length is the same. The alternative hypothesis would be the opposite, that they’re different. The significance level for this test will be 0.05. This hypothesis test will first be done with code created manually to show the step by step process, and then the built in t.test() function will be used.

First, the functions that will be used for our hypothesis test calculations are below. Functions for two sample degrees of freedom and a two sample test statistic will be used.

Degrees of freedom:

two_sample_df <- function(size1,size2,sd1,sd2, print = TRUE) {
  df_num <- ((sd1^2 / size1) + (sd2^2 / size2))^2
  df_den <- ((sd1^2 / size1)^2 / (size1-1)) + ((sd2^2 / size2)^2 / (size2-1))
  df <- floor(df_num / df_den)
  
  if (print == TRUE) {print(df)}
  
  return(df)
}

Test Statistic:

two_sample_test_statistic <- function(x_mean, y_mean, delta_null = 0, 
                                      sd1, sd2, size1, size2) {
  num <- x_mean - y_mean - delta_null
  den <- sqrt((sd1^2 / size1)+(sd2^2 / size2))
  return(num/den)
}

Next, the general statistics of each sample must be calculated. The mean, sample standard deviation, and sample size will be needed from each poet.

crane_mean <- mean(syllable_sample_one)
tomlinson_mean <- mean(syllable_sample_two)
crane_sd <- sd(syllable_sample_one)
tomlinson_sd <- sd(syllable_sample_two)
crane_n <- length(syllable_sample_one)
tomlinson_m <- length(syllable_sample_two)

After these values are collected, the test statistic can be calculated and a p-value will be generated from there.

alpha = .05
test_statistic <- two_sample_test_statistic(x_mean = crane_mean, y_mean = tomlinson_mean,
                                            sd1 = crane_sd, sd2 = tomlinson_sd, 
                                            size1 = crane_n, size2 = tomlinson_m)

degrees_of_freedom <- two_sample_df(size1 = crane_n, size2 = tomlinson_m,
                                    sd1 = crane_sd, sd2 = tomlinson_sd,
                                    print = FALSE)

p_value <- 2*pt(test_statistic, degrees_of_freedom, lower.tail = FALSE)

cat("The test statistic is:", test_statistic, 
    "\nThe degrees of freedom are:", degrees_of_freedom, 
    "\nThe p-value is:", p_value)

## The test statistic is: 4.791732 
## The degrees of freedom are: 54 
## The p-value is: 1.333705e-05

Based on the manual calculations done here, the generated p-value is less than .05 by a large margin. This would show that there is significant evidence that the two authors have a different true mean line length and that we should reject the null hypothesis.

t.test(syllable_sample_one, syllable_sample_two, paired=FALSE, alternative = "two.sided")

## 
##  Welch Two Sample t-test
## 
## data:  syllable_sample_one and syllable_sample_two
## t = 4.7917, df = 54.209, p-value = 1.325e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.609185 3.924149
## sample estimates:
## mean of x mean of y 
##  7.433333  4.666667

The built in t.test() provides near identical results, with only a slight variation in the p-value. This variation reaches the same conclusion though, that the null hypothesis should be rejected.

Conclusions

The results of this experiment line up with the authors assumptions entering into this project. As poetry is a very diverse and expressive form of art, it comes as no surprise that one would find significant evidence that the true average syllable lengths of the two authors would be different. The results of the hypothesis test back this up with a p-value that was far smaller than the chosen significance level.

The confidence interval for Stephen Crane had the true average of his poems line lengths being between 6.49 and 8.37 syllables. This makes sense as Crane frequently used 4 and 11 syllable lines, but primarily stuck to lines 7 and 8 syllables long which outweigh the outer values in the sample mean. What is surprising is the relative consistency the two poets had in their respective line lengths within their samples. Stephen Crane appeared to have three to four syllable lengths they used more frequently and Charles Tomlinson seemed to have one line length that was disproportionately represented over the others. Whether this is important or not is outside of the scope of this project but it does motivate a degree of reflection. The author wonders if other poets have a similar preference in their work. In retrospect, if this preference does exist and is not simply due to random chance it could just be a large part of a poets style. Another thing to consider is that different types of poetry lean more towards shorter or longer line lengths, so it’s possible there is more analysis that could be done on poets within a specific genre of poetry.