Type I error

Examples of Type I error in the following topics:

• Type I and II Errors

  • This type of error is called a Type I error.
  • The Type I error rate is affected by the α level: the lower the α level, the lower the Type I error rate.
  • It might seem that α is the probability of a Type I error.
  • If the null hypothesis is false, then it is impossible to make a Type I error.
  • Unlike a Type I error, a Type II error is not really an error.

• Outcomes and the Type I and the Type II Errors

  • The decision is to reject Ho when, in fact, Ho is true (incorrect decision known as a Type I error).
  • α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
  • The following are examples of Type I and Type II errors.
  • Type I error: The emergency crew thinks that the victim is dead when, in fact, the victim is alive.
  • The error with the greater consequence is the Type I error.

• Type I and Type II Errors

  • The two types of error are distinguished as type I error and type II error.
  • What we actually call type I or type II error depends directly on the null hypothesis, and negation of the null hypothesis causes type I and type II errors to switch roles.
  • An example of acceptable type I error is discussed below.
  • This is an example of type I error that is acceptable.
  • Distinguish between Type I and Type II error and discuss the consequences of each.

• References

  • D. (1979) Type I error rate of the chi square test of independence in r x c tables that have small expected frequencies.

• Summary of Formulas

  • α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
  • β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

• Elements of a Designed Study

  • The problem of comparing more than two means results from the increase in Type I error that occurs when statistical tests are used repeatedly.
  • Boole's inequality implies that if each test is performed to have type I error rate $\frac{\alpha}{n}$, the total error rate will not exceed $\alpha$.
  • These methods provide "strong" control against Type I error, in all conditions including a partially correct null hypothesis.
  • These methods have "weak" control of Type I error.
  • Discuss the increasing Type I error that accompanies comparisons of more than two means and the various methods of correcting this error.

• Student Learning Outcomes

• Multiple Comparisons of Means

  • ANOVA is useful in the multiple comparisons of means due to its reduction in the Type I error rate.
  • These errors are called false positives, or Type I errors.
  • Doing multiple two-sample $t$-tests would result in an increased chance of committing a Type I error.
  • where $T_i$ is the total of the observations in treatment $i$, $n_i$ is the number of observations in sample $i$ and CM is the correction of the mean:
  • The sum of squares of the error SSE is given by:

• Sample size and power exercises

  • (a) The standard error of ¯ x when s = 120 and (I) n = 25 or (II) n = 125.
  • (d) The probability of making a Type 2 error when the alternative hypothesis is true and the significance level is (I) 0.05 or (II) 0.10.
  • (b) Decreasing the significance level (α) will increase the probability of making a Type 1 error.
  • (d) If the alternative hypothesis is true, then the probability of making a Type 2 error and the power of a test add up to 1.
  • If the null hypothesis is harder to reject (lower α), then we are more likely to make a Type 2 error.

• Decision errors

  • What does a Type 1 Error represent in this context?
  • What does a Type 2 Error represent?
  • How could we reduce the Type 1 Error rate in US courts?
  • How could we reduce the Type 2 Error rate in US courts?
  • A Type 2 Error means the court failed to reject H 0 (i.e. failed to convict the person) when she was in fact guilty (H A true).