Day 10: STA205 - Introduction to Statistical Methods

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• Announcements:
  • There will be a short quiz at the end of the hour.

  • It's Valentine's Day!

    • You may not know the word "osculate" (to kiss). We use this word in math when we note, for example, that the normal curve "kisses" the x-axis as x moves far away from the mean.

      I found the word "osculate" used in this terribly interesting article about milk. I don't encounter the word much (outside of math):

      Sandra Steiger of the University of Bayreuth in Germany and her colleagues recently reported on an American species, Nicrophorus orbicollis -- a handsome, inchlong burying beetle with orange and black stripes -- in which both parents care for their young. A parent beetle will eat a bit of carrion, predigest it and, on being tapped on the mouth by an offspring's front legs, transfer the morsel into the little supplicant's mouth.

      "It's like kiss-feeding," Dr. Steiger said. "It looks really nice." But as the researchers demonstrated, there is more to the osculatory exchange than pulped meat: the parent's oral fluids are also critical to the young beetle's survival.

      Nothing says love like kiss-feeding...?:) I like "little supplicant", too: that's what I'm going to start calling my younger son, the college freshman!

    • Got this from the New York Times today: The Perfect Valentine? A Math Formula: Nothing says "I love you" like a customizable algebraic equation.

• Today we continue with Packet 04 (Hypothesis Test for p).
  • Science: we don't do truth, and we don't do proof: we test conjectures, hypotheses, theories. Sometimes they're about beer; sometimes they're about cola nuts!

  • Today we're going to be talking about errors in hypothesis testing, because sometimes we make mistakes while testing:

    Type I and Type II errors

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

    The following table (thanks Wikipedia) illustrates the situation in a typical medical context. Imagine that you're testing for AIDS. You either have it, or you don't. Now, suppose that you're tested for AIDS. The test either says you have it, or it doesn't. There are two kinds of errors the test can make:

    • You have AIDS, but the test says you don't.
    • You don't have AIDS, but the test says you do.
    That second error is bad, but the first one is worse!

    If the null hypothesis is that you have AIDS, the first error is called a Type I error, while the second error is a Type II:

    • Type I error: rejecting a true null hypothesis.
    • Type II error: failing to reject a false null hypothesis. (mnemonic: Type II, failing to)
    It's good to realize that we're getting two cases right!

    		Actual condition
    		Present	Absent
    Test  result	Positive	Condition Present + Positive result = True Positive	Condition absent + Positive result = False Positive Type I error
    	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.

  • Today we continue with Packet 04 (Hypothesis Test for p).

• Then, if we have a little more time, we'll begin 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 $p$ with given confidence.

• Links:
  • The Standard Normal Table of probabilities.

  • Statcrunch instructions:
    • Load the Stop and Frisk Data for packet 2. Load it by "WWW address", and make sure to set the delimiter to "comma" (because this is a "csv" (comma-separated values) file.
    • Once the data is loaded, click on "Graph" or "Table".
    • Alternatively, you may want to join my "group" on Statcrunch. That way you can access the data more quickly. It's Sta205-Long, if you want to add yourself into that group.
    • For the IMath homework, you'll need the Bipolar Depression Study data. Same thing: Load it by "WWW address", and make sure to set the delimiter to "comma" (because this is a "csv" (comma-separated values) file (or get it from the group).

• Today, if we have a little more time, we will startPacket 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 $p$ with given confidence.

  Let's remind ourselves about the distribution of $\hat{p}$:

  
  

  Now in this case we don't have $p$. We want to find $p$. So how do we proceed? That's the subject of this packet.

• Links:
  • Please watch Brooke's Hypothesis test for the proportion at some point.
  • The Standard Normal Table of probabilities.
  • Interview with Richard Feynman