Day 11: STA205 - Introduction to Statistical Methods

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Announcements:
- Quiz 6 - concerning using normals "backwards" -- will come at the end of the hour.
- We've got an exam coming up! Through 5.3, which we'll be wrapping up today...
Section 5.1: Sampling distributions and the normal curve (i.e., why is the normal distribution so important?)
- A statistic is a number that we calculate from a sample of data:
- A parameter is a number that could be calculated from a population if all of the data were accessible:
- Generally we pull many values from a population to form a sample.
- It turns out that if we take relatively large samples, sample after sample, and compute the statistic over and over, the histogram of statistic values, treated as a new sample, will look bell-shaped.
- The sampling distribution of the mean -- a different simulation
- Properties of the sampling distribution:
  - The mean of the sampling distribution of the sample mean , denoted , is equal to the mean of the population .
  - The standard deviation of the sampling distribution of the sample mean , denoted , is where n is the sample size. This statistic is called the standard error of the mean, and you will see it reported by StatCrunch, for example.
  - Here's the key thing: the sampling distribution is approximately normal for a large sample size n ("large" generally taken as greater than or equal to n=30).
  - See Figure 5.5, p. 172
  - If we're trying to estimate the population mean from the sample mean, we reason as follows:
    The sample mean is normally distributed about the population mean; so if we look at a given sample mean, it's not likely to be more than two standard deviations (or so) from the true population mean. Therefore the true mean should be about two standard deviations from the sample mean.
    Remember that "standard deviation" to the distribution of the sample mean is the standard error.
    If someone wants to challenge us, they can say "how big should my sample be to make sure that my sample mean is within x units of the true population mean?" We'll learn how to answer them.
    This is the key to Gallop polls, etc., when they say that the estimated statistics is "such-n-such plus or minus so-many percentage points". The size of the percentage points they want determines the sample size they must take.
- This result is encapsulated in The Central Limit Theorem (p. 167), and in the graphs of Figure 5.3, p. 168.
- Calculating probabilities for sample means involves computing Z-scores, as we've done in the past: see p. 171.
- Example problem: #4, p. 177
- Example 5.5, p. 191
- Exercise #6, p. 203
- Exercise #5, p. 202
Section 5.2: The Sampling distribution of the Sample Proportion
- Definition: Consider a sample of qualitative data for which one category, or attribute, is of interest. To describe such a sample, we will use the proportion of the sample having the attribute of interest. This statistic is denoted by the letter p.
- Properties of the sampling distribution of the sample proportion:
  - The mean of the sampling distribution of p, denoted , is equal to the mean of the population .
  - The standard deviation of the sampling distribution of p, denoted , is where n is the sample size.
  - Here's the key thing: the sampling distribution for p is approximately normal for a large sample size n ("large" generally taken as greater than or equal to n=30 at the very least, but in this case it depends as well on the value of ) as well.
  - Our choice of n to ensure normality is made so that three standard errors from the estimated are completely contained inside the interval [0,1]: and
  - See p. 186 for the strategy
  - See Figure 5.7 and 5.8, p. 182-183
- Calculating probabilities for p involves computing Z-scores, as usual
- Exercise #8, p. 189
- Exercise #5, p. 202

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