Lab 2: Data Types

Total Points: 40

Part 1. Review Questions (2+5+3 pts)

1. Multiply the inverse of a matrix \(\begin{bmatrix} 3 & 2 & 1\\ 4 & 8 & 1\\ 5 & 9 & 16 \end{bmatrix}\) with itself.

   Also, return those entries that are bigger than \(10^{-9}\).

# Your code here

2. Make a list lst1 with components

• 1:15 under the name num_vec;
• matrix(15:1, ncol = 3) under the name mat;
• rep(c("a", "x"), each = 3) under the name char_vec;
• list(x = c(1,2), y="STAT 302") under the name sublst.

Answer the following questions using R:

• Compute the sum of the component num_vec;
• What is the element in position [2,3] in the component mat?
• What is the third element in the component char_vec?
• Use the function strsplit() with argument split = "" to split the subcomponent y in the component sublst. What is the data type for the result strsplit()?
• Subset the result after strsplit() via [[1]]. What is the fifth element of this character vector?

# Your code here

3. Download the family.txt shown in Lecture 2 to your laptop. Then, read the file into R using the function read.delim(). Then, compute the following statistics in R:

• What is the standard deviation of ages in family.txt?
• What is the percentage of males in family.txt?
• What is the maximum BMI within all the female individuals?

# Your code here

Part 2: Normal Distribution (2+5+3+4 pts)

R provides several functions for the normal/Gaussian distribution:

• dnorm() computes the density function of a normal distribution;
• pnorm() calculates the percentiles (or equivalently, the cumulative distribution function) of a normal distribution;
• qnorm() returns the quantiles of a normal distribution;
• rnorm() generates the normally distributed random variables.

Use R to answer the following questions:

1. Create and store a vector norm_vec with \(100,000\) random variables from a Normal distribution with mean 6 and standard deviation 2. Print out the first 7 elements of norm_vec using the function head().

set.seed(123)  ## Don't change this line. It makes the result reproducible.
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2. Plot two histograms, one with the first 100 elements of norm_vec, and the other with all the elements of norm_vec. Set the argument freq = FALSE for both histograms for better comparisons.

• Change the x axis labels for both histograms to “Observations”.
• Set their titles as “Histogram of N(6,2) distributed random sample with n=THE CORRECT NUMBER OF SAMPLE POINTS”. Remember to change “THE CORRECT NUMBER OF SAMPLE POINTS”.
• Answer it by words: Which one looks more symmetric?

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3. Standardize the vector norm_vec to \(N(0,1)\) by subtracting its mean and then dividing it by its standard deviation. Name it as norm_vec_std. Compute the standard deviation of norm_vec_std. Also, what is the percentage of observations in norm_vec_std that are greater than 1.644854?

# Your code here

4. Apply the function pnorm() (without specifying any other arguments) to the vector norm_vec_std. Then, compute its mean and variance after applying the function pnorm(). Finally, plot its histogram after applying the function pnorm() with the argument freq = FALSE.

• Describe in words what do you see from the histogram. (Hint: How is the height of each bin compared with others?)

# Your code here

Part 3: Binomial Distribution (4pts per question)

The binomial distribution \(\mathrm{Bin}(m,p)\) is defined by the number of successes in \(m\) independent trials, each have probability \(p\) of success. Think of flipping an (unfair) coin \(m\) times, where the coin could be biased and has probability \(p\) of landing on heads.

Similar to the above normal distribution, R also provides several functions for the binomial distribution:

• dbinom() computes the probability mass function of a binomial distribution;
• pbinom() calculates the percentiles (or equivalently, the cumulative distribution function) of a binomial distribution;
• qbinom() returns the quantiles of a binomial distribution;
• rbinom() generates the random variables from a binomial distribution.

1. Initialize a matrix binom_mat with 3 columns and 100 rows, whose entries are all NA.

• Then, fill in each column with random samples from binomial distributions with \(m=300, p=0.25\) (first column), \(m=300, p=0.5\) (second column), and \(m=300, p=0.75\) (third column), respectively.
• Compute the column means of binom_mat.

set.seed(1234)  ## Don't change this line. It makes the result reproducible.
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2. Compute the means of every 10 elements in the first column of binom_mat. There should be 10 mean values in total. Then, output the median of these 10 mean values. Assign it to a variable MoM.

• Compared with the mean of the first column of binom_mat, is MoM closer to the expected mean 75? (Output a logical TRUE/FALSE using R!)

# Your code here

3. Now, change the first element in the first column of binom_mat to -100. Then, repeat what we did in Question 2 (i.e., compute the means of every 10 elements in the first column of binom_mat and then calculate the median as MoM2.)

• Now, compared with the mean of the first column of binom_mat, is MoM2 closer to the expected mean m*p = 75? (Output a logical TRUE/FALSE using R!)

# Your code here

4. Create a list binom_lst with 3 components:

• A vector with 500 elements from a \(\mathrm{Bin}(300,0.75)\) and name it as binom500;

• A vector with 1000 elements from a \(\mathrm{Bin}(300,0.75)\) and name it as binom1000;

• A vector with 20000 elements from a \(\mathrm{Bin}(300,0.75)\) and name it as binom20000.

• Compute the mean of each component of binom_lst. Which one is closest to the expected mean m*p = 225? Can you explain why?

• Look at the documentation of the functions qqnorm() and qqline(). Make QQ-plots with diagonal lines for each component of binom_lst. Which QQ-plot is most aligned with the diagonal line? Can you explain why?

set.seed(1234)  ## Don't change this line. It makes the result reproducible.
# Your code starts from here