Walter Jennings, a quantitative analyst with Smith, Kleen & Beetchnutty Brokerage, has just been informed of an important error in one of his recent statistical endeavors. Specifically, in one hypothesis test, Mr. Jennings failed to reject a null hypothesis that later was proven to be false. Which of the following best describes this type of error in hypothesis testing? Further, if the confidence level of the test were increased, would the probability of this error increase, decrease, or is this probability difficult to determine?
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A. B. C. D. E. F.A
In this example, Mr. Jennings has incorrectly failed to reject a null hypothesis. This type of error in hypothesis testing is called a Type II error. In hypothesis testing, the Type I error is given more attention than the Type II error. A Type I error is defined as the act of incorrectly rejecting the null hypothesis. In most hypothesis tests, the probability of a null hypothesis is equal to the significance level of the test. A significance level of 0.01, for example, indicates that a 1% chance exists that the null hypothesis will be rejected when it is indeed true. Another way to think of the probability of a Type I error is to observe the following relationship:
{Probability of a Type I error = (1 - confidence level)}.
For example, a confidence level of 95% leaves a 5% probability of a Type I error occurring. If this confidence level were to increase to say, 98%, then the probability of a Type I error would reduce to 2%. While a relationship exists between the confidence level for a hypothesis test and the probability of a Type I error, the relationship between confidence levels and Type II errors is not as explicit. The probability of a Type II error is inherently difficult to quantify, and as such, few hypothesis tests call for a determination of the probability of a Type II error. If the confidence level of a hypothesis test to be increased, it would be difficult to determine what effect, if any, this would have on the probability of a Type II error.