Day 19: STA205 - Introduction to Statistical Methods

Today:

Announcements:
- Quiz 10: Hypothesis testing returned
  1. Just two things needed: ${H_0:\mu=5}$
    ${H_a:\mu>5}$
  2. We aren't given ${\sigma}$ in this problem, so we can't actually do a test of hypothesis (yet!;). On the other hand, we can hope that s is a good approximation to ${\sigma}$ , so we can imagine that ${\sigma_{\bar{x}}\approx{s/\sqrt{n}}}\approx{.1}$ . Since 5 is about 9 standard errors from 5.9, we're pretty likely to conclude (using any confidence!) that the bags are heavier than 5 pounds on average.
We have an exam next time, which will be through 8.1.
I've posted all the old quizzes from this term.

Material to be covered for Exam 2:

Chapter 5:
- 5.2: pp. 188, 189, #1, 5
  - Sampling distribution of the sample proportion (p), mean and standard error (p. 181)
  - special rule for checking normality (proportions must fall between 0 and 1, p. 184)
  - Z-statistic for a proportion (p. 186)
- 5.3: pp. 201, 202, #1, 3
  - Making inferences about a conjectured value of ${\mu}$ or ${\pi}$ (p. 196)
- 5 Review:
  - Ch. 5 Review: pp. 205 - 207, #1, 2, 8
  - Summary: p. 205
Chapter 6:
- 6.1 pp. 221, 222, #2, 3, 4
  - Estimating ${\mu}$ via a Confidence interval (known ${\sigma}$ ) (p. 214)
  - Good choices for Z (90%, 95%, 99%) (p. 212)
  - Validity of the z-interval (p. 215)
  - Procedure to follow, p. 221
- 6.2 227 1, 3
  - Selecting the sample size (p. 224)
- 6 Review:
  - Ch. 6 Review 229 - 231 1, 3, 8
  - Summary, p. 229
Chapter 7:
- 7.1: pp. 242, 243, #3, 5
  - Elements of a test of a hypothesis (p. 234)
    - Null and alternative hypotheses
    - Observed significance level (p-value)
    - Test Statistic
    - Decision rule
    - Calculation
    - Interpretation
- 7.2: pp. 255, 256, #1, 3, 4
  - Performing a test of hypothesis about ${\mu}$ (p. 245, 246)
- 7.3: pp. 264, #1, 2
  - Possible errors (type I and type II)
- 7 Review:
  - Ch. 7 Review: pp. 269, 270, #2, 3, 4
  - Summary, p. 267
Chapter 8:
- 8.1: pp. 284 - 286, #3, 5, 7
  - Inferences about a population proportion ${\pi}$

Section 8.1: Inferences about a population proportion ${\pi}$

Confidence intervals: Essentially the same as for the sample mean, but
- the standard error formula is different: we don't have a given ${\sigma}$ , and hence a ${\sigma_p}$ , so we estimate it using the computed value ${s_p=\sqrt{\frac{p(1-p)}{n}}}$ based on the sample proportion p (see p. 273): the resulting confidence intervals are ${\left(p-z\sqrt{\frac{p(1-p)}{n}},p+z\sqrt{\frac{p(1-p)}{n}}\right)}$
- Example:
  - #6, p. 285
Hypothesis testing: Essentially the same as for the sample mean, except that you have a different formula for the z-statistic:

${z=\frac{p-\pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}}$
- Example:
  - #2, p. 284
Additional sample proportion example: #4, p. 285

Section 8.2: Estimating ${\mu}$ when ${\sigma}$ 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

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