Day 07: Statistics for Business Applications

• Areas of the histogram represent probabilities; similarly probabilities correspond to areas under the density curve (the bell-shaped curve).
• Calculating a Z-score (standardizing a normal curve. All normals are really the same - just shifted and stretched!)
• Using the Z table of normal probabilities (p. 235)

• Some examples (including using the table "backwards")
• Example: 8.44, p. 245
• Example: 8.58, p. 246

Reminder: in this section a variable is declared to be normally distributed, with a given mean and standard deviation. In this chapter, the assumption is that those are population values (parameter values); so those are "mu" and "sigma" (rather than xbar and s, the sample values)!

In reality, we very seldomly know that a variable is normally distributed, or the parameter values. Which brings us to Chapter 9....

• Section 9.1 reminds us simply that
  1. Parameters describe populations;
  2. Parameters are almost always unknown;
  3. We take random samples of a population as data, and
  4. we estimate parameter values from statistics calculated on the samples.

• Section 9.2: the sampling distribution of the mean

  Basic idea:

  • The variable of interest x is distributed with mean mu, with spread parameter (standard deviation) sigma; we want to estimate mu.
  • Instead of relying on a single value to estimate the parameter mu (the population mean), we use a sample of values, and take an average!
  • How is xbar distributed?
    • With the same mean mu,
    • with smaller standard deviation, and
    • more and more normally as the sample size gets big!
  We find that the larger the sample size, the closer is the sample mean sticks to the true mean (that is, the smaller the spread in the distribution of the means).

  Recall the dice applet - normal curves arise in the strangest places! (Figure 9.3, p. 273, for example)

• The sampling distribution of the sample mean (Central Limit Theorem)

• Some examples