Day 22: STA205 - Introduction to Statistical Methods

Today:

Announcements:
- Quiz over Hypothesis testing
- The tests have been graded:
  - Curve: Add 15 points to your score.
  - Furthermore, if you'll come see me, to discuss the exam and the points you've missed, I'll give you back 30% of the remaining points you missed. You should check out the key before coming to see me.
  - Key
  - If you do better on the final, I'll average it against the score of this test (or your worst test, whichever).
- What's in store for the rest of the course: three topics, namely
  - F-tests (comparing means of more than two populations)
  - Chi-square statistics
  - Linear regression
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

Section 9.3: Paired comparisons in estimating mean differences
- More good news: paired comparisons will turn out to be exactly like our first encounter with estimating ${\mu}$ in a single sample. So if you understood that, great!
- From our pairs of quantitative data, we create a new variable: the difference. So we form differences
  
  d₁=x₁-y₁,
  
  d₂=x₂-y₂,
  
  ...,
  
  d_n=x_n-y_n
  
  and then we work with this data, this random variable d.
- Notation:
  
  ${\mu_D}$ : Population mean
  
  ${\sigma_D}$ : Population Standard Deviation
  
  ${\bar{x}_D}$ : Sample Mean
  
  ${s_D}$ : Sample Standard Deviation
  
  ${n_D}$ : Sample size (number of pairs)
- Then Confidence Intervals and Hypothesis Testing are just as they were for single population samples (see pp. 346-347). One modest difference is that hypothesis tests are essentially always of the form ${H_0:\mu_D=0}$
- Examples (using StatCrunch):

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