04/14/2020

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- The following topics will be covered in this lecture:
- A quick discussion of confidence intervals for the variance
- Tests of significance
- The null hypothesis
- The alternative hypothesis
- The process of hypothesis testing
- Significance levels versus confidence levels
- Test statistics
- P-values
- Critical values
- Drawing conclusions
- Type I and type II errors

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- We have gone over in the last lecture how to estimate a
**population proportion \( p \)**and a**population mean \( \mu \)**. - In both cases, a
**sample statistic generates a “point estimate”**as a kind of “best guess” given a certain collection of data. - Likewise, we needed a
**“confidence interval”**to quantify**how uncertain**this best guess was, and to give**a range of other plausible values for the parameter**. - In both cases, our confidence interval
**needed to use some estimate of the standard deviation**of the**population**to**estimate our standard error**of the**sampling distribution**.

- the
**standard error**tells us how the**sample statistic**varies around the**true parameter****under replication of samples**. - When estimating the mean without knowledge of
**\( \sigma \)**, we used the sample standard deviation**\( s \)**to estimate the**standard error \( \sigma_\overline{x} \)**. - We know that
**\( s^2 \)**is the best, unbiased estimator for**\( \sigma^2 \)**, and although**\( s \)**is a biased estimator for**\( \sigma \)**, it is still usually the “best” option in some sense. - A more complicated question is the following,
**how do we produce confidence intervals for****\( \sigma \)**that take into account the uncertainty in our sample-based estimates of this parameter?- This is especially due to the fact that the
**sample variances**are**distributed right-skewed**around the**true population variance**.

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- Because this is a more complicated topic, and goes slightly beyond the overal scope of the course,
**this material will not have homework assignments or be tested**. - The purpose of the first part of this lecture is to give exposure to some advanced topics that will be useful for future work with statistical methods.
- The first advanced topic we will need to introduce is a very non-normal probability distribtuion, the
**“chi-square” distribution**. - Usually, this is denoted \( \chi^2(k) \) where the Greek letter chi denotes “chi-square”.

- The value
**\( k \) corresponds to the number of “degrees of freedom”**, like the student t degrees of freedom, and will be introduced shortly. - Before we introduce the \( \chi^2 \) distribution formally, we want to just note a few qualitative features of the distribution:
- If \( x \) is a random variable that behaves like \( \chi^2 \),
**\( x \) will only take on nonnegative values**, \[ x \geq 0 \] over any realization. - The distribution of values \( x \) under \( \chi^2 \) are
**right-skewed**, i.e., there are most values concentrated to the left around zero, but**many extremely large values that occur with much higher frequency than with a normal distribution**. - Interestingly, despite the differences from the normal distribution, \( \chi^2 \) is also closely related to the normal.