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IT 223 -- Jan 26, 2026

Review Exercises

1. Work problems from the Practice Problems on the Area under the Normal Curve.
   Answer: we worked problems 1s, 1e, and 2d in class today. The answers are shown on the page of practice problems.
2. Use R to create a data vector with entries from 1 to 100. Answer:
   

   v <- seq(1, 100, 1)
   # or
   v <- seq(1, 100)
   # or
   v <- 1:100
   

3. 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)
   

4. What is a z-score? How do you use R to compute the z-scores for a data vector?
   Ans: A z-score for individual observations is computed as z = (x - x) / SD.  It tells you how many
5. standard deviations the observation is away from the mean. The z-score can also be computed for the sample mean:
   z = (x - μ) / SD_ave. If you knew the population mean μ this z would tell you how many standard errors the sample mean was from the population mean. However, μ usually unknown, so this z can be used to obtain a confidence interval for μ.
   The z for individual observations is used to look up areas under the standard normal curve.

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

• We discussed this document last week.
• What proportion of observations are containted in each of these bins under the standard normal curve?
  

  [-1, 1]  [-2, 2]  [3, 3]
  

  Answer: 68% for [-1,1]; 95% for [-2,2]; 99.7% for [-3,3].

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

Normal Plots

We will finish discussing normal plots on Wednesday, Feb 28.

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

Practice Problems

We will discuss this section on Wednesday, Jan 28.

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 vector of 200 normal random numbers with μ=6.7 and σ=2.5 using the R rnorm function with arguments 200, 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 a vector of 200 uniform random numbers from the interval [1, 3.14) using the runif function with arguments 200, min=1 and max=3.14. Create the histogram and boxplot of this vector.

Project 2BCD

• Look at Project 2bcd.