Normal Distribution

Chapter 6 presented a discussion of probability distributions for discrete random variables.  This chapter presents a discussion of probability distributions for continuous random variables.  The approach to describing probabilities involving continuous random variables is different from that of discrete random variables.  With continuous random variables, students need to recognize the association between area and probabilities.

To help students understand the idea of using area to determine probabilities, I begin with a discussion of the uniform probability distribution.  The reason for this introduction is because the uniform distribution is arguably the easiest of the continuous distributions and students only need to find areas of rectangles to find probabilities. With the concept of areas representing probabilities in hand, I transition to the normal model.  The normal model is used to find probabilities, proportions, and percentiles.   The chapter also includes a discussion of normal probability plots to assess normality and concludes with a discussion of the normal approximation to the binomial distribution.

What to Emphasize

It is important that students understand that we cannot list all possible values of a continuous random variable, and therefore the idea of using tables to represent probability distributions is impossible.  So, we must represent the distribution of continuous random variables using mathematical formulas.  At this level, we graphically represent the distributions.  It is important that students associate areas with probabilities, proportions, and percentiles.

As you proceed through the chapter, continue to emphasize the interpretation of probabilities from a relative frequency point of view.

Section 7.1 Properties of the Normal Distribution -

This section provides an overview of the normal model. In particular, we discuss the properties of the model and focus on the use of area to represent a probability, proportion, or percentile.

  • The section begins with a discussion of finding probabilities of a uniform random variable.  The distribution is represented graphically (as a rectangle).  Since finding areas of rectangles by hand is easy and uniform probabilities are intuitive, it is not difficult for students to see the role area plays in finding probabilities.
  • With the uniform distribution as background, we transition to the normal model. Be sure students understand that the normal curve is a model meant to describe normal random variables.  Emphasize the interpretation of area as a proportion, probability, or percentile.
  • It may be helpful to use the normal applet so students can see the roles that μ and σ play in the shape of the normal curve.
Section 7.2 Applications of the Normal Distribution

This section discusses how to find area under the normal curve and how to find normal scores.

  • To find area under a normal curve, you may use either the tables or technology.  I recommend utilizing technology to find areas, but require students to draw normal models with the areas clearly labeled.
  • The section also presents a discussion of finding normal scores corresponding to an area.  Use technology to find normal scores.
  • Finally, I introduce z sub α notation.  This is important for the discussion of confidence intervals coming up in Chapter 9.
  • Classroom Example - Bone Age 
    • The ability to determine the age of some individuals can be difficult if there are not quality government records of birth. Bone growth takes place at the growth plates at the end of long bones.  Once all growth plates fuse, growth stops and an individual is consider a biological adult.  The age at which growth plates fuse for males is approximately normally distributed with a mean of 19 years with a standard deviation of 15.4 months.  Source: People smugglers, statisMtics and bone age.  Tim Cole  Significance Magazine June 2012 Volume 9 Issue 3(a) What is the probability a male’s growth plates fuse after age 20?
      (b) What is the probability a male’s growth plates fuse before age 18?
      (c) Determine the interquartile range for the age at which male’s growth plates fuse.
  • Classroom Example - Black Jack
    • Using a technique referred to as the “Hi-Opt I System” in the casino game black jack, a player can expect to earn $1.64736 per shoe of cards with a standard deviation of $48.92539 with a $1 bet per hand.  The distribution of earnings is normally distributed.   This assumes the shoe of cards contains 8 decks and the dealer reshuffles the shoe of cards when 80% of the cards have been dealt from the shoe.   Source: Hurley, W.J. and Pavlov, Andrey, “There Will Be Blood: On the Risk-Return Characteristics of a Blackjack Counting System”, Chance Vol. 24, No. 2, Spring, 2011
    • What is the probability of earning $10 from the shoe?
    • What is the probability of losing $10 from the shoe?
    • What is the probability of being “up” after one shoe?
Section 7.3 Assessing Normality

It is important to present the ideas behind normal probability plots.  This is especially true because much of inference on small samples requires that the sample data come from a population that is normally distributed.  However, constructing normal probability plots by hand is challenging, so we recommend using technology to draw the plots.

Section 7.4 The Normal Approximation to the Binomial Probability Distribution

The material on using the normal model to approximate binomial distributions is becoming less important with the advent of technology. Finding probabilities such as the probability of at least 400 successes in 500 trials of a binomial experiment without the use of technology would be time consuming, so using the normal approximation makes sense. However, with technology available, finding the exact probability using the binomial probability distribution function is quite easy.  Should you choose to present this topic, be sure to discuss the historical reasons for the material.  I do not cover this material in my course.