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IT 223 -- Project 2bcd: The Normal Distribution

Part b: Probabilities for Normal Distributions

• Use R to help you solve these problems. Show the R statements that you use to perform your calculations. Check your answers with the normal tables and extreme normal tables.
  Use the R function pnorm to find the area of the standard normal curve for the interval (-∞, z]. Use qnorm to find the z for which the area of (-∞, z] is q.
  
  1. LSAT scores are normally distributed with mean 150 and SD 10. What percentage of students taking the LSAT test obtain a score
     
     1. under 142?
     2. over 191?
     3. between 153 and 174?
  2. The lengths of horse pregnancies are normally distributed with mean 336 days and SD 3 days. What percent of the lengths of horse pregnancies are between 335 and 338 days?
  3. Determine the following percentiles for LSAT scores (see Problem 1 in this part):
          a. 57%      b. 98%      c. 39%
  4. Determine these percentiles for horse pregnancies:
     a. 89%      b. 45%

Part c: Random Number Simulations

• For this section include the two normal plots in your output, as well as the R statements that you use to create these plots.
• Perform two simulations with the R functions rnorm and runif to see how normal plots behave.
  
  1. Create a data vector consisting of 150 normal random numbers with μ= 5 and σ=3. Use the R function rnorm to create your dataset.
  2. Create a normal plot of the normal random numbers from Step 1. Describe this normal plot using one of these terms: approximately normal, skewed to the left, skewed to the right, thin tails, fat tails.
  3. Create a dataset consisting of 150 uniform random numbers, with minimum 2 and maximum 6. Use the R function runif.
  4. Create a normal plot of the uniform random numbers from Step 3. Describe this normal plot using one of these terms: approximately normal, skewed to the left, skewed to the right, thin tails, fat tails.

Part d: Checking if a Dataset is Unbiased and Homoscedastic

1. For your Project 2a dataset, (a) create a plot of the data points vs. observation number and (b) a normal plot. Is your dataset unbiased or biased? Is it homoscedastic or heteroscedastic? Is it approximately normally distributed? In each case, include a brief explanation.