Binomial Test

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What is a Binomial Test?

Binomial Test

  • A binomial test uses sample data to evaluate a Hypothesis about the values of p and q for a population consisting of binomial data.
  • The measurement scale consists of precisely two categories
  • Each individual observation in a sample is classified into only one of the two categories
  • Sample data consist of the frequency or number of individuals in each category

Prerequisite

  • Binomial distribution
  • z-score hypothesis tests
  • Chi-square test for goodness of fi

Hypothesis Test

1) Just Chance

  • The hypothesis states that two outcomes in the population would be predicted simply by chance.

H0 : p = q

2) No Change or No Difference from a Known Population

  • Null hypothesis: Proportions for one population are not different from the Know proportions of the existing population.

Example: drivers having a license

Binomial Test

Test Statistic for the Binomial Test

  • When the values pn ≥ 10 and qn ≥ 10 , binomial distribution approximates a normal distribution.
  • The shape of the distribution is approximately normal.
  • The mean of the distribution is µ = pn
  • The standard deviation of the distribution is  Binomial Test
  • the z-score corresponding to each value of X in the binomial distribution.

Binomial Test

Binomial Test

Binomial Test

Decision

  • If the z-score for the sample data is in the critical region, Reject H0 and conclude that the discrepancy between the sample proportions and the hypothesized population proportions is significantly greater than chance.
  • If the z-score is not in the critical region, fail to reject Ho.

Sign Test

  • The test is applicable for a repeated-measures study that compares two conditions, it is often possible to use a binomial test to evaluate the results.
  • Measuring each individual in two different treatment conditions or at two different points in time.
  • Two possible directions are coded by signs, with a positive sign indicating an increase and a negative sign indicating a decrease. When the binomial test is applied to signed data, it is called a sign test.

Null hypothesis

There is no difference between the two treatment conditions being compared.

H0 = p (increase) = q (decrease)

Critical region

With pn ≥ 10 and qn ≥ 10 🡺 Normal approximation to the binomial distribution is appropriate.

With α = .05, the critical region consists of z-scores greater with a sample size n

Test statistic calculation

Binomial Test

Decision

  • If data is in the critical region, we reject H0 and conclude that there is a significant effect
  • If data is outside the critical region, we accept H0 and conclude that there is no significant effect.

Zero Differences in the Sign Test

  • The null hypothesis in the sign test refers only to those individuals who show some difference between treatment 1 vs. treatment 2.
  • The null hypothesis does not consider individuals with zero difference between the two treatments.
  • Individuals who show no difference actually support the null hypothesis and should not be discarded.
  • Therefore, an alternative approach to the sign test is to divide individuals who show zero differences equally between the positive and negative categories. (With an odd number of zero differences, discard one, and divide the rest evenly.)
  • This alternative results in a more conservative test; that is, the test is more likely to fail to reject the null hypothesis
  •  

When to Use the Sign Test

  • When you have infinite or undetermined scores, a t-test is impossible, and the sign test is appropriate.
  • It is possible to describe the difference between two treatment conditions without precisely measuring a score.
  • In this situation, the data are sufficient for a sign test, but you could not compute a t statistic without individual scores
  • Often a sign test is done as a preliminary check on an experiment before serious statistical analysis begins.
  • Occasionally, the difference between treatments is not consistent across participants. This can create a very large variance for the different scores. However, the sign test only considers the direction of each difference score and is not influenced by the variance of the scores.

Binomial Test vs Chi-Square

  • The binomial test evaluates the same primary Hypothesis as the chi-square test for goodness of fit.
  • Both tests evaluate how well the sample proportions fit a hypothesis about the population proportions.
  • The relationship between the two tests can be expressed by the equation

χ2 = z2

    • -χ2 is the statistic from the chi-square test for goodness of fit
    • -z-score from the binomial test.

Refer to the Engineering Statistics Handbook

 

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