Normal probability distributions part III and introduction to estimation

03/31/2020

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Outline

  • The following topics will be covered in this lecture:
    • Assessing normality of data
    • Histograms
    • Q-Q plots
    • Examples in StatCrunch
    • Point estimates for population proportions
    • Confidence intervals for population proportions
    • Critical values again

Motivation

Diagram of the percent of outcomes contained within each standard deviation of the mean for a normal distribution.

Courtesy of Melikamp CC via Wikimedia Commons

  • Recall now the bell curve picture that we often consider – we will suppose we have a population that is distributed as a bell shape.
  • We suppose that the population mean is \( \mu \) and population standard deviation \( \sigma \).
  • Normally distributed data is characterized by the following features:
    1. The frequencies start low, then increase to one or two high frequencies, and then decrease to a low frequency.
    2. The distribution is approximately symmetric.
    3. There are few if any extreme values.
  • These features can be understood as a direct application of the empirical rule.
  • We suppose that the histogram represents the sample data which is mostly bell-shaped, but the collection is smaller than the population so it is not exact.
    • In particular, any data set is subject to sampling error and we cannot expect a perfect bell shape from a small sample even when the population is perfectly normally distributed.
  • So far, we have used histograms to examine these features in data and to assess normality.
    • However, histograms sometimes have technical issues that make them unreliable.

Motivation continued

Histogram of IQ scores.

Courtesy of Mario Triola, Essentials of Statistics, 5th edition

  • Let’s consider a histogram we saw earlier with IQ scores.
    • Over the population of US adults, IQ scores are normally distributed, exhibiting the bell shape discussed earlier.
    • However, depending on the choice of bin-widths for the histogram, we can see a very different shape.
    • Here the bin-widths are quite wide and so we collect many scores together – this obscures the bell shape.
    • On the other extreme, if we chose bins so finely that every bin had one observation, the histogram would be flat.
    • With a “good” choice in between, we can see the bell shape more clearly.