Day 18: STA205 - Introduction to Statistical Methods

Today:

Announcements:
- Quiz 9: confidence intervals (returned).
  - When you give an interval, put the small number first: e.g. (1,3) rather than (3,1).
  - In the formula ${(\bar{x}-z\left(\frac{\sigma}{\sqrt{n}}\right),\bar{x}+z\left(\frac{\sigma}{\sqrt{n}}\right))}$ There are two things we can affect: n and z. They are independent. Increasing one does not change the other.
    Some people claimed that, because we were increasing confidence, the sample size n is increasing. That's one way to increase confidence in an interval of fixed size: if ${\frac{\sigma}{\sqrt{n}}}$ stays fixed, while n increases, then z must go up (meaning that confidence is increasing).
    The other way to increase confidence that you've trapped the true value is to give a bigger interval (increase z).
- Quiz 10: Hypothesis testing
- That quiz reminds me of "Beergate"....

Section 7.3: Possible errors and level of significance

Reality versus Decision (sometimes we make mistakes): Type I and Type II errors

Alternative vocabulary: false positive, false negative....

The following table (thanks Wikipedia) illustrates the situation:

		Actual condition
		Present	Absent
Test result	Positive	Condition Present + Positive result = True Positive	Condition absent + Positive result = False Positive Type I error
Test result	Negative	Condition present + Negative result = False (invalid) Negative Type II error	Condition absent + Negative result = True (accurate) Negative

The probability of a Type I error is given by ${\alpha}$ , the level of significance of the test.
The probability of a Type II error is given by ${\beta}$ .
Obviously we'd like both types of errors to be small, but making one smaller tends to make the other larger! There's always a trade-off.
Examples:
- #3, p. 265
- #4, p. 265

Section 8.1: Inferences about a population proportion
- 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).
  - 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 when is unknown
- 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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