Lecture Notes

Chapter 8 Hypothesis Sampling and Effect Size: One-sample tests (Revised: 2003-07-30)

• The Use of Hypothesis Testing

  Much of inferential statistics consists of hypothesis testing. Science is often described as a process of observation, subsequent hypothesis generation, and hypothesis testing. So, what is a hypothesis? A hypothesis is an informed guess or a hunch, but based upon previous observation.

  For example, suppose you observe that every time you feed your dog some particularly delectable food, the dog tucks her tail down. You may generate some hypotheses from this observation: the dog wants to hide her scent from other dogs in order to eat all of the food herself, or the more delicious she finds the food the greater the degree of tail-tucking. After hypotheses are generated they must be tested. If they are not tested you have not done science.

  Obviously, not all of your hypotheses will pan out. Those that do not pan out must be discarded, a painful but necessary process. Those hypotheses that do pan out are kept and become part of the knowledge base of science. Another part of hypothesis testing consists of investigating whether your observations and subsequent hypotheses apply to populations. In other words, do all dogs perform the same behavior under the same circumstances?

• The Logic of Hypothesis Testing
  • "My spouse is not cheating on me." That statement could be a kind of null hypothesis. So you hire a private detective (i.e., you run an experiment), and you discover infidelity, so you must reject the null hypothesis. Your spouse is cheating on you.
  • Suppose, however, that your private detective cannot find any evidence of infidelity. Is you spouse now proved faithful?
  • The two alternatives above illustrate the logic of hypothesis testing nicely. Rejection of the null hypothesis means that it is false.
  • The null hypothesis is a way of setting up hypotheses so that you assume that there is no expected difference. Usually, you seek to establish a difference, and you do so by rejecting the null hypothesis. Sometimes, however, you may wish to accept the null hypothesis (i.e., to show that two groups are the same).
  • Symbolic statement of null hypothesis:
    • H₀: m₁ = m₂

      where m₁ and m₂ are population means

  • An alternative hypothesis is used to possibly account for the true state of affairs. In the example above, the alternative hypothesis would be: "My spouse is cheating on me."
  • Should the null hypothesis be rejected, then the alternative hypothesis might better describe the true state of affairs.
  • Symbolic statements of the alternative hypotheses (Note: use two-tailed for H₁ and one-tailed for H₂ and H₃)
    • H₁: m₁ <> m₂
    • H₂: m₁ > m₂
    • H₃: m₁ < m₂
  • Use hypothesis testing:
    • to know a parameter that cannot be measured (i.e., you must use sampling)
    • to test whether two sample means come from the same population (the classic experiment)
• Alpha Levels
  • a is usually set at 0.05 or 0.01
  • a is the criterion for which you will determine statistical significance
  • The rejection region consists of those values of t at the ends of the distribution
    • these are Table D values that exceed the t value obtained
  • The t table provides you with the critical values necessary to make such a decision
    • Obtain these values by selecting an a level and the appropriate degrees of freedom
• Type I Errors

  Type I errors occur when a null hypothesis is rejected but it should not have been. In other words, the two or more distributions are not significantly different, but you said they were. Type I errors are sometimes called a false acceptance. Note that setting your significance level higher will lower your chances of making a type I error. Example:

  Null hypothesis--rail crossing lights are flashing and train is not coming.

  Your behavior--you stop anyway

  Consequence--you lose a little time

  Type I error--false acceptance of crossing signal

• Type II Errors

  Type II errors occur when a null hypothesis is NOT rejected when it should have been. In other words, the two or more distributions are significantly different, but you said they were not. Type II errors are sometimes called false rejectance. Note that setting your significance lower will lower your chances of making a type II error.

  Null hypothesis--rail crossing lights are flashing and train is coming.

  Your behavior--you do not stop

  Consequence--you lose life

  Type II error--false rejectance of crossing signal

• Type I And II Errors In Practice

  In practice, most researchers attempt to not make type I errors. The logic being that it is safer to falsely accept the null hypothesis, thus missing a difference between distributions, than to falsely reject the null hypothesis and promote a difference in distributions that does not really exist. Also, realize that you may not raise the probability of making fewer type I and type II errors at the same time. As one type goes up, the other must come down. Can you see why? Here is a discussion of Type I and Type II errors using the legal system as an example--http://www.intuitor.com/statistics/T1T2Errors.html

• One Tail or Two?
  • Most of the time, we will use the two-tailed test
  • If you are only looking for the extremes of a distribution, you may use one-tailed test
    • Typically, one-tailed is used in some applied situations
  • Note that you will set your alpha level higher if you use one-tailed test when two-tailed is called for
• Using the t distribution
  • One sample t test
  
  • Sample Problem (Number 20, page 173)

  The extroversion population mean is 50. Thirteen car sales persons took the test and their mean score was 56.10, and their standard deviation was 10. What was the result of a one-sample t test? What does it mean?

  Solution: use the formula for one-sample t test

  

  or: 56.10 - 50.00 / 2.774 = 2.199; df = 12

  t _.05(12df) = 2.179, so 2.199 exceeds the t value and we may reject the null hypothesis

• Effect Size
  • The idea here is to estimate the size of the difference of the two means in question.
  • We will not go into the more advanced issue of power analysis, but will learn to compute an effect size index, d
  

  Evaluate d using the following levels:

  Small effect	d = .20
  Medium effect	d = .50
  Large effect	d = .80

• Using t to Test r
  • The t test can be used to test the null hypothesis of no difference between the r of a bivariate set of data (remember however, that you still cannot infer causality, all you are doing is determining what role chance has in the correlation obtained)
  • r is the symbol for the correlation coefficient of a population, and you are testing the null hypothesis:
  H₀: r = .00 (H₁: r <> .00 is the alternative hypothesis)
  • use the formula:

• Sample problem (Number 25 (b), p. 176

for an r = -.19, with N = 122, decide if the null hypothesis can be rejected for that correlation.

• Solution:

Using the formula above, find that t = -2.12 with df = 120, the table value of t_.05(120) = 1.98. Therefore, we may reject the null hypothesis, and the correlation is not likely to have occurred by chance.

• URLs
  • The Logic of Hypothesis Testing
    • The Null Hypothesis (1 of 4)--(Rice) (works-2003-07-30)
    • Why the Null Hypothesis is not Accepted (1 of 5)--(Rice) (works-2003-07-30)
    • Type I and Type II errors--interesting discussion using the legal system to explain error types (works-2003-07-30)
      • http://www.intuitor.com/statistics/T1T2Errors.html
• QUIZ