Question 1(ii) : Plot a histogram of the means of the samples generated in part (i), using an
appropriate option in R for plotting the histogram on the probability density
The task is to plot a histogram of the means of the samples generated in part (i) using R, and the histogram should be plotted on the probability density scale. Here’s the code and an explanation of the solution:
# Load the previously generated samples load("exp_samples.RData") # Calculate the means of each group of samples sample_means <- tapply(samples, rep(1:num_samples, each = sample_size), mean) # Plot the histogram on the probability density scale hist(sample_means, main = "Histogram of Sample Means (Exp(3) Distribution)", xlab = "Sample Means", ylab = "Probability Density", prob = TRUE, # Set prob = TRUE for probability density scale col = "skyblue", # Color of the bars border = "black", # Color of the bar borders breaks = 30) # Number of histogram bins
Explanation of the code and solution:
- Load the Previously Generated Samples:
load("exp_samples.RData")loads the previously generated samples from the “exp_samples.RData” file. This step is necessary because we need the data for further analysis and visualization.
- Calculate Sample Means:
sample_means <- tapply(samples, rep(1:num_samples, each = sample_size), mean)calculates the means of each group of samples. The
tapplyfunction is used to apply the
meanfunction to each group of samples. The
repfunction is used to create a vector that repeats the sample numbers (from 1 to the number of samples) for each group. This gives us a vector of sample means.
- Plot the Histogram:
hist(sample_means, ...)plots the histogram of the sample means.
ylabare used to add a title and labels to the plot.
prob = TRUEis the key option for plotting the histogram on the probability density scale. When
probis set to
TRUE, the heights of the histogram bars are scaled so that the total area under the histogram equals 1, making it a probability density histogram.
borderspecify the colors of the bars and their borders, respectively.
breaksspecifies the number of histogram bins.
In summary, the code loads the previously generated sample data, calculates the sample means, and then plots a histogram of the sample means. The key aspect of this histogram is that it’s plotted on the probability density scale, meaning that the area under the histogram represents probabilities, making it suitable for visualizing the distribution of sample means from the Exponential distribution.
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