Day 11: STA205 - Introduction to Statistical Methods

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• Announcements:
  • Reminder: please hide your rectangles.

  • Your quiz is returned.
    • population: KY adults; sampling frame: KY adults registered to vote.
    • check $np > 10$, $n(1-p) > 10$, and random sample
    • $H_0: p=.286$; $H_a: p<.286$
    • Type I: reject a true null. $p=.285$, but we mistakenly conclude that $p<.286$.

  • Your first exam is next time -- this Thursday.

    • The exam will cover everything through packet 5.

    • Questions will resemble those of IMath homework, and the packets. Study both to prepare yourself.

    • You'll need only a calculator and writing utensil (maybe a couple, of different colors). That's it. (You may not use a cell for a calculator.) Make sure that you have a calculator -- and know how to use it.

    • You'll receive a standard normal table and formula sheet.

    • Last time I asked you to Come prepared with questions for Tuesday. Did you? I'll do a short review of the material to be covered, and then open up the floor for questions.

• Last time:
  • We nearly wrapped up Packet 5, which has two major objectives:
    1. estimate parameter $p$ using statistic $\hat{p}$, with given confidence, and
    2. discover the sample size needed to estimate $p$ to a given tolerance with given confidence.

    So we were learning how to calculate sample size to achieve a certain sized confidence interval.

• Today
  • A brief overview, and then your questions.

  • Some key ideas from each section:

    1. Packet 1:

       1. Normal, bell-shaped curves are controlled by just two numbers -- a mean, and a standard deviation.
          

       2. All normal curves are fundamentally the same -- and can be "standardized" to the Z-table, where observations $y$ are turned into $z$ values by $z=\frac{y-\mu}{\sigma}$.

       3. The Empirical rule gives areas under the curve for a few steps of the standard deviation.

       4. The Z-table allows us to compute areas more finely: left tails, right tails, and between values.

       5. Areas correspond to probabilities.

    2. Packet 2:

       1. Statistics is the science of reasoning from data: descriptively or inferentially.

       2. Population (of "observational units")

       3. Sampling Frame (that part of the population accessible to our sampling)

       4. Various sampling schemes: simple random, convenience, voluntary

       5. We collect data on a sample -- quantitative or categorical -- and then compute statistics on the sample which (we hope) reflect the values of parameters of the population.

       6. We talked about box-plots to illustrate frequencies of responses -- how do we represent data?

    3. Packet 3: Sampling distribution of sample proportion $\hat{p}$

       1. The sample proportion $\hat{p}$ is computed from a categorical variable -- the fraction of the population having that characteristic of interest.

       2. We're hoping that the distribution of $\hat{p}$ is normal -- if the sample is random, and $np>10$ and $n(1-p)>10$.

          If so, the normal distribution of $\hat{p}$ has mean $p$, and standard deviation ${\sqrt{\frac{p(1-p)}{n}}}$.

       3. And if so, we can answer some questions about $p$ from samples.

    4. Packet 4: Hypothesis test for population proportion $p$

       1. Provided we can assume normality of $\hat{p}$, then we can perform an hypothesis test for $p$.

       2. Someone proposes a value for $p$, and we have other ideas (an alternative hypothesis). The idea is to test the null hypothesis $p=...$ against our alternative.

       3. We take a sample, compute a $\hat{p}$, and ask "how likely is that value of $\hat{p}$, given an assumption that the mean of the distribution of $\hat{p}$ is $p$?

       4. We compute the probability of a value of \[ z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}} \] this extreme or more so from our one sample.

       5. If it is less than the significance level $\alpha$ (chosen before the test), then we "reject the null hypothesis $H_0$ in favor of the alternative $H_a$."

       6. There are two kinds of errors we can make:

    5. Packet 5: Confidence intervals and sample size

       1. Provided we can assume normality of $\hat{p}$, then the confidence interval (CI) \[ C.I.=\hat{p} \pm z_{crit}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

       2. $z_{crit}$ is the value of $z$ corresponding to the confidence desired.

          The more confidence desired, the larger the CI required (so it's harder for the true value to "leak out").

          The larger the sample size, the smaller the CI for given confidence.

       3. We can calculate the sample size required for a given margin of error (MOE) and confidence: \[ n=\left(\frac{z_{crit}}{MOE}\right)^2 p (1-p) \]

          Since we don't know $p$ (that's what we're after), we have a choice -- we can use a good estimate for $p$, or use the conservative value $p=0.5$ -- that leads to largest value of $n$.

          But this large value of $n$ assures us that we achieve the $MOE$ required.

• Links:

  • Type I and Type II errors

  • Sampling Distribution of the Sample Proportion Calculator

  • The Standard Normal Table of probabilities.