Binomial Distribution

Learning Objectives

  1. Describe the logic by which it can be concluded that someone can distinguish between two things
  2. State whether random assignment ensures that all uncontrolled sources of variation will be equal
  3. Define precisely what the probability is that is computed to reach the conclusion that a difference is not due to chance
  4. Distinguish between the probability of an event and the probability of a state of the world
  5. Define "null hypothesis"
  6. Be able to determine the null hypothesis from a description of an experiment
  7. Define "alternative hypothesis"
  8. The Null Hypothesis

    The hypothesis that an apparent effect is due to chance is called the null hypothesis. In the Physicians' Reactions example, the null hypothesis is that in the population of physicians, the mean time expected to be spent with obese patients is equal to the mean time expected to be spent with average-weight patients. This null hypothesis can be written as:

    μobese = μaverage

    or as

    μobese - μaverage = 0.

    The null hypothesis in a correlational study of the relationship between high-school grades and college grades would typically be that the population correlation is 0. This can be written as

    ρ = 0

    where ρ is the population correlation (not to be confused with r, the correlation in the sample).

    The probability value is the probability of an outcome and not the probability of a particular state of the world. In statistics, it is conventional to refer to possible states of the world as hypotheses since they are hypothesized states of the world. Using this terminology, the probability value is the probability of an outcome given the hypothesis. It is not the probability of the hypothesis given the outcome.

    Although the null hypothesis is usually that the value of a parameter is 0, there are occasions on which the null hypothesis is a value other than 0. For example, if one were testing whether a subject differed from chance in their ability to determine whether a flipped coin would come up heads or tails, the null hypothesis would be that π = 0.5.

    Keep in mind that the null hypothesis is typically the opposite of the researcher's hypothesis. In the Physicians' Reactions study, the researchers hypothesized that physicians would expect to spend less time with obese patients. The null hypothesis that the two types of patients are treated identically is put forward with the hope that it can be discredited and therefore rejected. If the null hypothesis were true, a difference as large or larger than the sample difference of 6.7 minutes would be very unlikely to occur. Therefore, the researchers rejected the null hypothesis of no difference and concluded that in the population, physicians intend to spend less time with obese patients.

    If the null hypothesis is rejected, then the alternative to the null hypothesis (called the alternative hypothesis) is accepted. The alternative hypothesis is simply the reverse of the null hypothesis. If the null hypothesis

    μobese = μaverage

    is rejected, then there are two alternatives:

    μobese < μaverage

    μobese > μaverage.

    Naturally, the direction of the sample means determines which alternative is adopted. Some textbooks have incorrectly argued that rejecting the null hypothesis that two populations means are equal does not justify a conclusion about which population mean is larger.