Day 21: STA205 - Introduction to Statistical Methods

Today:

Announcements:
- Sorry, the tests aren't graded yet (I'll have them done on Thursday).
- No quiz today, but they may recommence on Thursday.
Section 8.2: Estimating when is unknown
- We finally get real! We don't know ${\sigma}$ in general. So we're going to be estimating it....
- The t-distribution
  - similar in shape to the normal
  - we estimate ${\sigma}$ using the sample standard deviation, s:
    
    ${z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}}$ whereas ${t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}}$
  - Properties of the t-distribution above:
    - symmetric, like the normal
    - centered at zero, like the normal
    - had n-1 degrees of freedom
    - the standard deviation of the distribution is ${\sqrt{\frac{df}{df-2}}}$
    - Since the standard deviation of the distribution is greater than 1, it is a little broader than the normal, a little more variable.
  - The major difference between the z- and t-distributions is that for the normal we have one single table; for the t-distribution, we need to know the value of n, and then use the table for that value of df=n-1. (Rats!) Of course, with technology, this is not really much of a problem....
- How does this affect our strategy? It's basically the same: see page 290....
- Let's try an example:
  - #3, p. 303
  - #5, p. 304
Section 9.1: Inferences comparing two populations
- Our over-riding objective is to decide whether two populations share a parameter or not: e.g. is ${\mu}$ the same for both?
- There are two kinds of samples of the populations we'll compare:
  - Dependent (paired) samples
  - Independent (unpaired) samples
- Dependent (paired) samples:
  - husband versus wife hours in the kitchen
  - male and female graduates matched on major and GPA
  - "pre" and "post" intervention scores
- Independent (unpaired) samples:
  - 100 students randomly chosen from sta205 at NKU versus 100 students randomly chosen from the equivalent of sta205 at EKU
  - 1000 voters randomly chosen in KY versus 900 voters randomly chosen in OH.
- One thing that separates the two pretty clearly is when the sample sizes are different: it's almost a sure sign that the samples are independent.
- Examples:
  - #3, p. 328
  - #5, p. 328

Section 9.2: Inferences about two population means: independent samples

Notation:

Population 1	Mean: ${\mu_1}$	Variance: ${\sigma^2_1}$
Sample 1	Mean: ${\bar{x}_1}$	Variance: ${s^2_1}$	Sample size: ${n_1}$
Population 2	Mean: ${\mu_2}$	Variance: ${\sigma^2_2}$
Sample 2	Mean: ${\bar{x}_2}$	Variance: ${s^2_2}$	Sample size: ${n_2}$

We're going to be making inferences about ${\mu_1-\mu_2}$ :
Conditions under which we may do so (p. 330):
- Two independent random samples must be selected
- both sample means must be normally distributed
Standard error of ${\bar{x}_1-\bar{x}_2}$ ${\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}$
Confidence interval: usual story:
point estimate ${\bar{x}_1-\bar{x}_2}$ plus or minus t times the standard error:

${(\bar{x}_1-\bar{x}_2)-t\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}$ to ${(\bar{x}_1-\bar{x}_2)+t\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}$

where the degrees of freedom are really horrible (p. 331) -- use technology!
Test of hypothesis, usual story:
${H_0:\mu_1-\mu_2=0}$
${H_a:\mu_1-\mu_2<0,\mu_1-\mu_2>0}$ or ${\mu_1-\mu_2\ne{0}}$ .
Decision rule: Accept H_a if p-value ${<\alpha}$
Test Statistic: ${t=\frac{(\bar{x}_1-\bar{x}_2)-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}}$ with appropriate degrees of freedom (p. 331 -- use technology!)
Examples (using StatCrunch):
- For the following exercises, do not "pool" the variances (that is done when there is reasonable suspicion that the variances of the two samples is the same -- we have no such suspicions)
- Example 9.3, p. 332
- #6, p. 341

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