To Notes

IT 223 -- Jan 28, 2026

Review Exercises

1. What is the standard normal density?
2. IQ scores are normally distributed with mean = 100 and SD=15. How many persons out of 100 have an IQ score greater than 120?
3. IQ scores are normally distributed with mean = 100 and SD = 15. How many persons out of one billion have an IQ score greater than 175? Use the Extreme Values of the Normal Distribution table.
4. What is the 90th percentile for IQ scores. mean = 100 and SD = 15 for IQ scores.
5. What are the definitions of unbiased, biased, homoscedastic, and heteroscedastic.
6. Set up the R vector x with values from 1 to 200 and the R vector y that contains 200 standard normal random values. Then create separate plots for y1, y2, y3, and y4 all vs. x for these R definitions:
   

   y1 <- y
   y2 <- y * (x / 100)
   y3 <- y + (x - 100) * 0.3
   y4 <- y * (x / 100) + (x - 100) * 0.3
   

   Classify each plot as unbiased or biased; homoscedastic or heteroscedastic.

Normal Plots

• Normal plots can be used to determine if a dataset is approximately normal, or how a dataset deviates from normality.
• We compare the actual data points on the y-axis with the expected normal scores, which divide the standard normal density into into n+1 equal areas.
• In a normal plot, the sorted actual data points are plotted on the y-axis and the corresponding expected normal scores are plotted on the x-axis.
• Note: there are at least nine different ways to form a normal plot. The default R algorithm is slightly different than simply plotting data points vs. expected normal scores. However, the normal plot looks basically the same no matter which algorithm that is used.

Practice Problems

1. Compute normal scores for a dataset of size 9.
   Answer: Choose the z-scores that divide the standard normal curve into 9 + 1 = 10 equal areas:
   -1.28  -0.84  -0.52  -0.25  0.00  0.25  0.52  0.84  1.28
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. Answer:
   

   > x <- c(81, 95, 97, 101, 112, 125, 129, 167, 220)
   > qqnorm(x)
   

   
   

R Practice Problems

1. Create 50 values of a normal random variable x with μ = 15, σ = 3.8.
   
   1. Create a histogram of x.
   2. Create a normal plot of x.
   Answer:
   

   > x <- rnorm(50, mean=15, sd=3.8)
   > hist(x)
   > qqnorm(x)
   

2. Create 50 values of a uniform random variable x in the range [10, 50]:
   

   > x <- runif(50, min=10, max=50)
   

   1. Create a histogram of x.
   2. Create a normal plot of x.
   Answer:
   

   > x <- runif(50, min=10, max=50)
   > hist(x)
   > qqnorm(x)
   

Bivariate Datasets

• Practice Problem: Look at the Bears 2024 Roster: bears-2024-roster.txt. Use R to plot the weight in kilos vs. the height in meters for each player. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.

Correlation

• Practice Problem: With the Bears Roster 2024 Dataset, use R to compute the correlation of the height in meters vs. the weight in meters for each player. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.

Linear Regression

• Practice Problem: With the Bears Roster 2024 Dataset, use R to:
  
  1. Compute the height in meters and the weight in kilos of the Bears players. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.
  2. Find the simple linear regression equation for predicting weight in kilos from height in meters.
  3. Create the residual plot, which is the residuals (y-axis) vs. the predicted values (x-axis)

Project 3

• Look at the project descriptions for Project 3.