A manager of the credit department for an oil company would like to determine whether the mean monthly balance of credit card holders is equal to $102. An auditor selects a random sample of 32 accounts and finds that the mean owed is $111.5 with a sample standard deviation of $23.15. Suppose you wanted to conduct a test to determine whether the auditor should conclude that there is evidence that the mean balance is different from $102. In the context of this questions, which of the following interpretations of Type II Error is correct?[Fill in the blank]

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BlackTom AI · Accepted Answer

First, let's restate the scenario to anchor the ideas: a hypothesis test is being conducted to see if the mean balance differs from 102. Type II error occurs when we fail to reject the null hypothesis even though the alternative is true.
Option 1: 'We conclude the mean monthly balance is greater than $102 when in fact it is not' — This describes concluding a difference in a specific direction (greater) when there isn't one. That is a Type I error (false positive in a directional test) rather than a Type II error.
Option 2: 'We conclude the mean monthly balance is not different from $102 when in fact it is' — This matches the textbook definition of a Type II error: failing to detect a real difference (the null is false, but we do not reject it). It does not specify a direction of the difference, just that a difference exists and was not detected.
Option 3: 'We conclude the mean monthly balance is different from $102 when in fact it is not' — This states that we detect a difference when there actually isn’t one. That is a Type I error (false positive), not Type II.
Option 4: 'We conclude the mean monthly balance is not greater than $102 when in fact it is' — This is a directional miss, but it describes failing to reject in a specific directional sense when the true difference is in the opposite or any non-null direction. It still aligns more with a mis-specified directional error rather than the standard Type II framing, and it’s not the canonical Type II error definition.
Option 5: 'We conclude the mean monthly balance is different from $102 when in fact it is' — This says we detect a difference when there is one, which is a correct rejection of a false null (i.e., the power scenario). This is not a Type II error; it corresponds to a successful detection (a correct decision or, at least, a correct rejection).
Thus, among the choices, the interpretation that matches Type II error is the one where we fail to detect a real difference by not rejecting the null when the alternative is true: Option 2. The other options describe Type I errors, directional errors, or correct rejections, which are not Type II errors.

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