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IT 223 -- Apr 17, 2024

Review Exercises

1. Use R to create a data vector with entries from 1 to 100. Answer:
   

   > v <- 1:100
   

2. Use R to create a data vector with entries that start at 1, end at 3, and increase by 0.01 from one entry to the next? Answer:
   

   > v <- seq(1, 3, 0.01)
   

3. What is the difference between SD and SD+? Answer:
   SD = square root of (sum of squared deviations divided by n)
   SD+ = (sum of squared deviations divided by n-1)
   SD+ is slightly larger than SD.
4. Which R functions do you use to do compute following for the data vector x:
   
   1. Sample mean: mean(x)
   2. Sample standard deviation (SD+): sd(x)
   3. Sample median: quantile(x, 0.5) or median(x)
   4. Histogram: hist(x)
   5. Boxplot: boxplot(x)
   6. Scatterplot of data vector vs. observation number: plot(1:length(x), x)
5. Use the R function plot to draw a standard normal density histogram, for which μ=0 and σ=1. Use the xlim argument to set the range of x to [-4, 4]. Use the ylim argument to set the range of y to [0, 1]. Use type="l" to connect the heights of the histogram with line segements. Use the main parameter to set the title of the plot. Answer:
   

   > x <- seq(-4, 4, 0.05)
   > y <- dnorm(y)
   > plot(x, y, type="l", xlim=c(-4, 4), ylim=c(0, 1), main="Standard Normal Density")
   

6. What is a z-score? How do you use R to compute the z-scores for a data vector?
   Answer: a z-score for an observation tells you how many standard deviations away from the sample mean the observation is.  You compute the z-scores of a data vector like this:
   

   z = (x - mean(x)) / sd(x)
   

The Normal Distribution

• The following areas under the standard normal curve come up often; commit them to memory:
  
  1. Area [-1, 1] = 0.68 = 68%
  2. Area [-2, 2] = 0.95 = 95%
  3. Area [-3, 3] = 0.997 = 99.7%

Some R functions:
dnorm, pnorm, qnorm, rnorm

• dnorm(x, mean, sd) -- The height of a normal density with μ and σ specified by the mean and sd arguments.
  Example: draw a plot of the normal density of SAT scores, where μ=1500 and σ=300. Use seq(0, 2500, 1) for the x-values. Set the title of the plot (main argument) to "Density of SAT Scores". Answer:
  

  > x <- seq(0, 2500, 10)
  > y <- dnorm(x, mean=1500, sd=300)
  > plot(x, y, type="l", main="Density of SAT Scores")
  

• pnorm(x, mean, sd) -- The area under the normal density above the interval (-∞, x].
  Example: if μ=1500 and σ=300 for SAT scores, find the proportion of scores that are greater than 1950. of Answer:
  

  > pnorm(1950, 1500, 300)
  [1] 0.9331928
  

  However, we don't want area (-∞, 1950], we want area (1950, ∞) = 1 - area (-∞, 1950]. The answer we want is
  

  > 1 - pnorm(1950, 1500, 300)
  [1] 0.0668072
  

• qnorm(p, mean, sd) -- The p quantile for the normal density.
  Example: If μ=1500 and σ=300 for SAT scores, what is the 0.95 quantile or 95th percentile for SAT scores?
  Answer:
  

  > qnorm(0.95, 1500, 300)
  [1] 1993.456
  

• rnorm(n, mean, sd) -- Generate n normally distributed random numbers with the specified mean and SD.
  Example: Generate 100 normally distributed random values with mean=1500 and sd=300. Then create the histogram, the box plot, and the plot of the x-values vs observation number.
  

  > x <- 1:100
  > y <- rnorm(x, 1500, 300)
  > hist(y)
  > boxplot(y)
  > plot(x, y)
  

Biased vs. Unbiased; Heteroscedastic vs. Homoscedastic for Graphs

• The following scatterplots are plots of x_i (measurement) vs. i (observation number) with the sample mean marked with a red horizontal line. The measurement is plotted on the vertical axis; the observation number is plotted on the horizontal axis. What does each plot tell you? Describe each plot using these terms:
  
  • Unbiased   The average of the observations in every thin vertical strip is the same all the way across the scatterplot.
  • Biased   The average of the observations changes, depending on which thin vertical strip you pick.
  • Homoscedastic   The variation (SD+) of the observations is the same in every thin vertical strip all the way across the scatterplot.
  • Heteroscedastic   The variation (SD+) of the observations in a thin vertical strip changes, depending on which vertical strip you pick.
     
    
    
    Ans: (a) unbiased and homoscedastic, (b) unbiased and heteroscedastic, (c) biased and homoscedastic, (d) biased and heteroscedastic, (e) unbiased and heteroscedastic, (f) biased and homoscedastic.

Practice Problems

1. Work problems from the Practice Problems on the Area under the Normal Curve.

Normal Plots

• Normal plots can be used to determine if a dataset is approximately normal, or how a dataset deviates from normality.

Practice Problems

1. Compute normal scores (Van der Waerden's method) for a dataset of size 9.
2. Construct the normal plots by hand of this dataset:
    
          81   95   97   101   112   125   129   167   220
3. Create the normal plot for this dataset with R.

Random Variable Simulation

• A random variable is the process of choosing a random number.
• R can generate many different types of random numbers, including from a normal distribution with a specified μ and σ.
• Example 1:   Generate 200 normal random numbers using the R rnorm function with mean=6.7 and sd= 2.5. Create the histogram and boxplot of these random numbers
• A uniform random variable with range [a,b) is a value drawn from the interval [a,b); every value in this interval is equally likely to be chosen.
• Example 2:   Generate 200 normal random numbers from the interval [1, 3.14). Create the histogram and boxplot of these numbers.

Project 2BC

• Look at Project 2BC.