Day 14: STA205 - Introduction to Statistical Methods

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Today:

Announcements:
- We begin with a short quiz....
Section 5.3: Using conjectured values of the population mean to make inferences:
- Check out p. 196 for strategy, first; then
- Exercise #5, p. 202
Chapter 6 (section 6.1 and 6.2)
- First of all, this chapter deals only with ${\bar{x}}$ , sampled from a population with known ${\sigma}$ and unknown ${\mu}$ .
- What do we know about ?
- We have two major objectives:
  1. estimate ${\mu}$ , with given confidence (section 6.1);
  2. find out what the sample size should be to estimate ${\mu}$ , with given precision (section 6.2).
- How do we go about achieving these objectives?
  - Estimating with given confidence (Section 6.1)
    - What does it mean to estimate something "with confidence"?
    - "confidence interval" -- what should that mean?
    - z-interval for ${\mu}$ (see figure on p. 211, then formula on p. 214)
    - Here's a table of common choices for z:
      
      Desired Confidence 90% 95% 99% ???
      
      z-value 1.645 1.960 2.576 get it from the z-table!
    - See p. 221 for conditions under which we may do this.
    - Example (follow Strategy of p. 216):
      - #1, p. 221
      - What if we needed only a 90% confidence interval?
      - What if we wanted a 99% confidence interval?
  - On to Objective 2 (Section 6.2):
    - What if we require a 95% confidence interval with a known precision?
      That is, we want to know that ${\mu}$ lies within a certain interval ${(\bar{x}-E,\bar{x}+E)}$ with 95% confidence.
    - Then we will have to design our experiment well! In particular we will have to select an appropriate sample size.
    - We know that the larger the sample size, the "tighter" the normal curve of the distribution of ${\bar{x}}$ .
    - Hence n determines how tight the curve needs to be, by the equation ${n=\left(\frac{z\sigma}{E}\right)^2}$ , where the value of z corresponds to the desired confidence (in this case, 95%), and E corresponds to the "half-width" of the interval.
    - Example:
      - #4, p. 227
      - Notice that it's not necessary to know the mean value in order to calculate the sample size! Only the margin of error E, z, and the known value of ${\sigma}$ .

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Desired Confidence	90%	95%	99%	???
z-value	1.645	1.960	2.576	get it from the z-table!