Estimating the standard deviation and introduction to hypothesis testing part I



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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

Confidence intervals for the variance

Sample variaces are distributed right-skewed around the true population parameter.

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.

Confidence intervals for the variance continued

Chi square distribution is skewed.

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:
    1. If \( x \) is a random variable that behaves like \( \chi^2 \), \( x \) will only take on nonnegative values, \[ x \geq 0 \] over any realization.
    2. 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.

Chi-square distribution