Day 15: STA205 - Introduction to Statistical Methods

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Today:

Announcements:
- Return of Quiz 7
  1. We must be clear on the difference between the population and a sample.
  2. Under what conditions may we assume normality?
Chapter 6 (section 6.1 and 6.2)
- 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).
- 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)
    
    ${(\bar{x}-z\left(\frac{\sigma}{\sqrt{n}}\right),\bar{x}+z\left(\frac{\sigma}{\sqrt{n}}\right))}$
  - 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.
  - Let's look at the simulation on p. 217.
  - 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 ${\sigma}$ .
Chapter 7: Testing conjectures about when the value of is known
- By an hypothesis, we will mean a conjecture about the nature of a population.
- A test of hypothesis is a statistical procedure used to make a decision about the conjectured value of a population parameter.
- Please refer to your Hypothesis testing handout
- Example from our first test: Problem 5 can be phrased in this terminology. We were "testing a conjecture", but we can formalize the process as a test of hypothesis (see p. 240).
  1. Null hypothesis: ${\mu = 265}$
  2. Alternative hypothesis: ${\mu > 265}$
  3. Decision rule: ${p\le{.05}}$ , then reject the null in favor of the alternative.
  4. Test statistic: z-statistic, ${Z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}}$
  5. Compute p-value (value as extreme as Z=2.08 or more so): ${p = .0188}$ , hence reject the null in favor of the alternative.
  6. Interpretation of the results: older folks apparantly listen to more radio than other adults (our estimate is 290 hours per year).
- Examples:
  - #1, p. 242 (proportion)
  - From the course website of worked examples
    Warning: I don't like the wording of this example: honestly, we don't accept hypotheses: we either
    
    reject them in favor of the alternative, or
    fail to reject them.

Links

Website maintained by Andy Long. Comments appreciated.

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