To Lecture Notes

IT 223 -- 1/19/11

Review Questions

1. What does it mean to say that the normal distribution is ubiquitous?

   Ans: It means that it appears all over the place; it is hard not to find it showing up in everyday life applications.

2. Who first published the equation for the normal histogram?
   
   p(x) = (1 / √2π) e^-0.5x2 = 0.39894 * 2.71828^-0.52

   Ans: Abraham DeMoivre in 1756.

3. Give the definition of the standard error of the mean?

   Ans: SD_ave = SD / √n

4. What does it tell you?

   Ans: It give you an idea of how accurate the sample mean is for estimating the true measurement in an ideal measurement model. You divide by square root of n because the mean is a more accurate measure of μ than an individual observation.

5. What are the original and current definitions of the meter, the second, and the kilogram?

   Ans: See the discussion in The Ideal Measurement Model.

The Normal Distribution

• The normal distribution is ubiquitous in statistics.

• Since the sample mean is defined as
  x = x₁ / n + ... + x_n / n,

  it is the sum of independent random variables. Because of the Central Limit Theorem, x is approximately normally distributed if n is large enough.

• Not only is often x approximately normally distributed if an experiment is repeated many times, many other random variables are normally distributed, thanks to the Central Limit Theorem.

• Here is a discussion of The Normal Distribution.

Practice Problems

1. Work these problems from the Practice Problems on the Area under the Normal Curve.
   2 b, d.

2. For a population of giraffes, the heights are measured. The sample mean is 16.5 ft; the sample standard deviation is 3.8. What percentage of giraffes are taller than 20 ft.

3. What is the 93rd percentile for the giraffes in Practice Problem 2?

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:
   1. 61   66   69   77   88   92   97

   2. 81   95   97   101   112   125   129   167   220

3. Create normal plots from the datasets in Problem 2 using SPSS.

Random Variable Simulation

• A random variable is the process of choosing a random number.

• SPSS can generate many different types of random numbers, including from a normal histogram with a specified μ and σ.

• Example 1:   Generate 20 normal random numbers with μ = 6.7 and σ = 2.5.
  1. Create a column with variable name i. Enter the values 1 to 20.

  2. Create a new variable, by setting the target variable to x and the expression to RV.NORMAL(6.7, 2.5).

• 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 20 normal random numbers from the interval [1, 3.14).
  1. Create a column with variable name i. Enter the values 1 to 20.

  2. Create a new variable, by setting the target variable to y and the expression to RV.UNIFORM(1, 3.14).

SPSS Practice Problems

1. Create 50 values of a normal random variable x with μ = 15, σ = 3.8. You may wish to use Excel to create a column of numbers from 1 to 50, then copy and paste into SPSS.

2. Create a histogram of x with superimposed normal curve.

3. Create a normal plot of x.

4. Create 50 values of a uniform random variable y in the range [10, 50).

5. Create a histogram of y with superimposed normal curve.

6. Create a normal plot of y.

Project 2

• Look at Project 2.

• Verify that you know how to create normal plots and how to perform the simulations in Part B.

Bivariate Datasets and Correlation

• This document discusses Bivariate Datasets and Correlation.