sheet03.R

### Stats with R Exercise Sheet 3

##########################
#Week4: Hypothesis Testing
##########################

## This exercise sheet contains the exercises that you will need to complete and 
## submit by 23:55 on Monday, November 18th. Write the code below the questions. 
## If you need to provide a written answer, comment this out using a hashtag (#). 
## Submit your homework via moodle.
## You are required to work together in groups of three students, but everybody 
## needs to submit the group version of the homework via moodle individually.
## You need to provide a serious attempt at solving each exercise in order to have
## the assignment graded as complete. 


## Please write below your (and all of your teammates') name, matriculation number. 
## Name: 1. H T M A Riyadh, 2. Abdallah Bashir, 3. Maria Francis
## Matriculation number: 1. 2577735, 2. 2577831, 3. 2573627

## Change the name of the file by adding your matriculation numbers
## (exercise0N_firstID_secondID_thirdID.R)


###############
### Exercise 1: Deriving sampling distributions
###############
## In this exercise, we're going to derive sampling distributions with 
## different sizes.

## a) Load the package languageR. We're going to work with the dataset 'dative'. 
## Look at the help and summary for this dataset.
library(languageR)
#?dative
summary(dative)

## The term dative alternation is used to refer to the alternation between 
## a prepositional indirect-object construction
## (The girl gave milk (NP) to the cat (PP)) and a double-object construction 
## (The girl gave the cat (NP) milk (NP)).
## The variable 'LenghtOfTheme' codes the number of words comprising the theme.

## b) Create a contingency table of 'LenghtOfTheme' using table(). 
##    What does this table show you?
table(dative$LengthOfTheme)

## c) Look at the distribution of 'LenghtOfTheme' by plotting a histogram and a boxplot. 
##    Do there appear to be outliers? Is the data skewed?

hist(dative$LengthOfTheme)
boxplot(dative$LengthOfTheme)
# there are lot of outliers appeared. the data is skewed positively

## d) Now we're going to derive sampling distributions of means for different 
##    sample sizes. 
##    What's the difference between a distribution and a sampling distribution?

#Distribution is process where every single observation of the population (whole population)is spreaded up and plotted
# but Sampling distribution is the distribution of the sample of the population.
#  


## e) We are going to need a random sample of the variable 'LengthOfTheme'. 
##    First create a random sample of 5 numbers using sample(). 
##    Assign the outcome to 'randomsampleoflengths'

randomsampleoflengths <- sample(dative$LengthOfTheme, size = 5)

## f) Do this again, but assign the outcome to 'randomsampleoflengths2'. 

randomsampleoflengths2 <- sample(dative$LengthOfTheme, size = 5)

## g) Now calculate the mean of both vectors, and combine these means 
##    into another vector called 'means5'.

mean1 <- mean(randomsampleoflengths)
mean2 <- mean(randomsampleoflengths2)
means5 <- mean1 + mean2

## h) In order to draw a distribution of such a sample, we want means of 
##    1000 samples. However, we don't want to repeat question e and f 
##    1000 times. We can do this in an easier way: 
##    by using a for-loop. See dataCamp or the course books for 
##    how to write loops in R.


for(var in 1:1000){
  randomsampleoflengths <- sample(dative$LengthOfTheme, size = 5)
  randomsampleoflengths2 <- sample(dative$LengthOfTheme, size = 5)
  x_mean <- mean(randomsampleoflengths)
  y_mean <- mean(randomsampleoflengths2)
  mean_sum <- x_mean + y_mean
  means5 <- c(means5 , mean_sum)
}

  

## i) Repeat the for-loop in question h, but use a sample size of 50. 
##    Assign this to 'means50' instead of 'means5'.

for(var in 1:50){
  randomsampleoflengths <- sample(dative$LengthOfTheme, size = 5)
  randomsampleoflengths2 <- sample(dative$LengthOfTheme, size = 5)
  x_mean <- mean(randomsampleoflengths)
  y_mean <- mean(randomsampleoflengths2)
  mean_sum <- x_mean + y_mean
  means50 <- c(mean50 , mean_sum)
}
  
## j) Explain in your own words what 'means5' and 'means50' now contain. 
##    How do they differ?
# means5:- contains 1000 observations of sumation of the mean of two random sample where each sample 
# contains 5 variables 

# means50:- contains 50 observations of sumation of the mean of two random sample where each sample 
# contains 5 variables

#they differ from each other because means5 contains 1000 observation and means50 contains only 50.
#more observations always gives better result then less observation.

## k) Look at the histograms for means5 and means50. Set the number of breaks to 15.
##    Does means5 have a positive or negative skew?
hist(means5, breaks = 15)
#means5 have a positive or negative skew

hist(means50, breaks = 15)

## l) What causes this skew? In other words, why does means5 have bigger 
##    maximum numbers than means50?

#It causes skwe because one of its tail is bigger then other.
#means5 has 1000(*2 = 2000) random observation (high frequency) but means50 has only 50(*2 = 100) 
#random observation. More frequency gives the maximum numbers.


###############
### Exercise 2: Confidence interval
###############

## A confidence interval is a range of values that is likely to contain an 
## unknown population parameter.
## The population parameter is what we're trying to find out. 
## Navarro discusses this in more depth in chapter 10.

## a) What does a confidence interval mean from the perspective of experiment replication?


#The confidence interval tells you where the true mean of a population lies with a certain level of certainty.
#You can say how big the interval is, for example say you want to find the interval in which the true mean of 
#the population will be in with a probability of 95% (5% chance that the true mean is outside this interval)


## b) Let's calculate the confidence interval for our means from the previous 
##    exercise.
##    First, install and load the packages 'lsr' and 'sciplot'

library(lsr)
library(sciplot)


## c) Look at the description of the function ciMean to see which arguments it takes.

?ciMean()

## d) Use ciMean to calculate the confidence interval of the dataset dative from
##    the previous exercise.
##    Also calculate the mean for the variable LengthOfTheme.

ciMean(dative)
#LengthOfRecipient   [1.771194 1.913146]
#LengthOfTheme       [4.121943 4.421115]
mean(dative$LengthOfTheme)
#4.271529

## e) Does the mean of the sample fall within the obtained interval? 
##    What does this mean?

#The sample mean does fall within the obtained interval. This only means that
#the mean is within 1.96 standard deviations of the true mean of the population.


## f) As the description of dative mentions, the dataset describes the 
##    realization of the dative as NP or PP in two corpora.
##    The dative case is a grammatical case used in some languages 
##    (like German) to indicate the noun to which something is given.
##    This dataset shows us, among other things, how often the theme is 
##    animate (AnimacyOfTheme) and how long the theme is (LengthOfTheme).
##    Plot this using the function bargraph.CI(). Look at the help for this function. 
##    Use the arguments 'x.factor' and 'response'.

bargraph.CI(x.factor = dative$AnimacyOfTheme, response = dative$LengthOfTheme)

## g) Expand the plot from question f with the ci.fun argument 
##    (this argument takes 'ciMean'). 
##    Why does the ci differ in this new plot compared to the previous plot?

bargraph.CI(x.factor = dative$AnimacyOfTheme, response = dative$LengthOfTheme, ci.fun = ciMean)



###############
### Exercise 3: Plotting graphs using ggplot.
###############
# There are many ways of making graphs in R, and each has their own advantages 
# and disadvantages. One popular package for making plots is ggplot2. 
# The graphs produced with ggplot2 look professional and the code is quite easy 
# to manipulate.
# In this exercise, we'll plot a few graphs with ggplot2 to show its functionalities.
# You'll find all the information you'll need about plotting with ggplot2 here: 
# http://www.cookbook-r.com/Graphs/
# Also, you have been assigned the ggplot2 course in DataCamp. Please work through 
# this course.

## a) First install and load the ggplot2 package. Look at the help for ggplot.
install.packages("ggplot2")
library(ggplot2)
## b) We're going to be plotting data from the dataframe 'ratings' 
##    (included in languageR). 
##    Look at the description of the dataset and the summary.
library(languageR)
data = ratings
str(data)
summary(data)
## For each word, we have three ratings (averaged over subjects), one for the 
## weight of the word's referent, one for its size, and one for the words' 
## subjective familiarity. Class is a factor specifying whether the word's 
## referent is an animal or a plant. 
## Furthermore, we have variables specifying various linguistic properties, 
## such as word's frequency, its length in letters, the number of synsets 
## (synonym sets) in which it is listed in WordNet [Miller, 1990], its 
## morphological family size (the number of complex words in which 
## the word occurs as a constituent), and its derivational entropy (an 
## information theoretic variant of the family size measure). 
## Don't worry, you don't have to know what all this means yet in order to 
## be able to plot it in this exercise!

## c) Let's look at the relationship between the class of words and the length. 
##    In order to plot this, we need a dataframe with the means.
##    Below you'll find the code to create a new dataframe based on the existing 
##    dataset ratings.
##    Plot a barplot of ratings.2 using ggplot. Map the two classes to two 
##    different colours. 
##    Remove the legend.
summary(ratings)
condition <- c("animal", "plant")
frequency <- c(mean(subset(ratings, Class == "animal")$Frequency), mean(subset(ratings, Class == "plant")$Frequency))
length <- c(mean(subset(ratings, Class == "animal")$Length), mean(subset(ratings, Class == "plant")$Length))
ratings.2 <- data.frame(condition, frequency, length)
str(ratings.2)


ggplot(ratings.2, aes(y = length,x= frequency,color = condition)) + geom_bar(stat="identity", fill="white")

## d) Let's assume that we have additional data on the ratings of words. 
##    This data divides the conditions up into exotic and common animals 
##    and plants.
##    Below you'll find the code to update the dataframe with this additional data.
##    Draw a line graph with multiple lines to show the relationship between 
##    the frequency of the animals and plants and their occurrence.
##    Map occurrence to different point shapes and increase the size 
##    of these point shapes.
condition <- c("animal", "plant")
frequency <- c(7.4328978, 3.5864538)
length <- c(5.15678625, 7.81536584)
ratings.add <- data.frame(condition, frequency, length)
ratings.3 <- rbind(ratings.2, ratings.add)
occurrence <- c("common", "common", "exotic", "exotic")
ratings.3 <- cbind(ratings.3, occurrence)
ratings.3
ggplot(ratings.3, aes(x=  frequency, y= occurrence, group= condition,color =condition)) + geom_line()



## e) Based on the graph you produced in question d, 
##    what can you conclude about how frequently 
##    people talk about plants versus animals, 
##    with regards to how common they are?
## people tend to speak about exotic animals more frequently than common animals, 
## on the other hand, most frequent plants that people speak about are the common plants.