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The Standard Normal Probability Distribution

Project: The History of Distributions

When students are able to place what they are leaning in mathematics in historical context, the material becomes much more interesting. Students like to hear stories about people and places. The following research topics will help them relate to the mathematics and mathematicians. The information they gather can be written up in a report, presented to the class with visual aids like PowerPoint, or both.

Assignment One:

Each student, or group of students, should be assigned one of the distributions studied in class, and research the following topics.

  • What mathematician(s) or statistician(s) developed the distribution? Give a short biography of their life or lives.
  • Describe the time period when the distribution was developed. What historic events happened around that time? What other scientific or mathematical discoveries where made?
  • How was the distribution discovered? Was their a particular need or topic of inquiry that lead to its discovery?
  • How is the distribution related to other distributions?

Assignment Two:

There are an amazing number of different distributions in use. Students could also find a completely new distribution to research. In this case they can include the following topics as well as the ones mentioned above.

  • What are the characteristics and parameters of the distribution?
  • Give some examples of situations where the distribution is used to calculate probabilities.
  • What is the shape of the distribution? Display graphs of the distribution with different values of the parameter(s).

The Density Curve of the Normal Distribution

Project: Analyzing Normally Distributed Data (with Excel)

This project makes use of important concepts and skills that have been developed in the previous chapters. It gives the students a chance to bring many topics together, use technology, and work with real data.

Procedure:

  1. Think of a data set that is approximately normally distributed and for which you can gather at least 30 elements.
  2. Enter the data into a spreadsheet and calculate the mean and standard deviation of the data set.
  3. Calculate the z-score for each element of the data set and plot the z-score against the data values.
  4. Make a histogram of the data. Use small bin widths so the histogram appears somewhat smooth.
  5. Draw in an approximation of the density curve. Mark the mean and inflection points.
  6. Calculate the percent of the data that lies within one standard deviation of the mean. Repeat the process for two and three standard deviations.

Analysis: Address the following topics in writing.

  • Describe the shape of the distribution. Hopefully it is approximately normally distributed, but describe where it deviates from the idealized normal curve. Is it a bit skewed? Are there any outliers? Where is the symmetry off? ...
  • Use the normal probability plot to analyze how well the distribution approximates a normal distribution.
  • Are the inflection points of the distributions located one standard deviation away from the mean? Use the data to explain any discrepancies.
  • Compare the percents calculated in step six to the Empirical Rule. Use the data to explain any discrepancies.

Applications of the Normal Distribution

Explore: Area Under a Continuous Distribution

The probability of attaining a certain value of a discrete random variable is equal to the area of the rectangle over that value in the probability distribution. With a continuous distribution, an interval is substituted for the discrete value. This creates an interesting anomaly when finding the probability of one exact value. Take example four in the text concerning the height of twelve year old boys in Britton. What is the probability of randomly selecting a boy with height exactly 155\;\mathrm{cm}? Here the exactly is taken very seriously. It is 155\;\mathrm{cm}, not 154.99999999\;\mathrm{cm} or 155.0001\;\mathrm{cm}. This probability would correspond to an area with height given by the normal density function and width zero, so the probability would be zero. There must be some interval or range of acceptable heights in order to calculate a probability other than zero.

Procedure:

  1. Use the normal distribution to find the probability of randomly selecting a boy between 154.9\;\mathrm{cm} and 155.1\;\mathrm{cm}. Here we are allowing a tolerance of 0.1\;\mathrm{cm}.
  2. Use the normal distribution to find the probability of randomly selecting a boy between 145.5\;\mathrm{cm} and 155.5\;\mathrm{cm}. Her we are allowing a tolerance of 0.5\;\mathrm{cm}.
  3. Use the normpdf function on your calculator to find the height of the normal density function at x = 155\;\mathrm{cm}. Compare this result with your answer in #2. Explain this relationship.

Answers:

  1. 0.009
  2. 0.045
  3. 0.046, The numbers are extremely close since the width of the area is one. The slope of the normal curve is not decreasing at a constant rate though, so the probabilities are not exactly the same.

The interval becomes more important when using integrals to calculate probabilities with other continuous distributions in more advanced probability classes. Similar modifications are made when using the normal approximation to the binomial distribution with the correction for continuity. Introducing student to the concept now will give them an advantage later.

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