Understanding Type 1 Errors in Hypothesis Testing Statistics Problems

Step 1: Understand the problem statement.
The problem describes a scenario involving a hypothesis test about the battery life of new cell phone batteries. The claim is that the new batteries last longer than the typical cell phone battery life, which is nine hours.
The null hypothesis ( ) is that the mean battery life ( ) is equal to 9 hours.
The alternative hypothesis ( ) is that the mean battery life ( ) is greater than 9 hours.
We are asked to identify what constitutes a Type 1 error in this scenario.

Step 2: Define a Type 1 error.
A Type 1 error occurs when we reject the null hypothesis ( ) when it is actually true. In other words, we conclude that there is a significant effect or difference when, in reality, there is none.

Step 3: Relate the hypotheses to the scenario.
In this case:
(The average battery life is 9 hours).
(The average battery life is greater than 9 hours).

A Type 1 error means we reject and conclude that is true, even though is actually true.
So, a Type 1 error is concluding that the average battery life is greater than 9 hours ( ) when, in reality, the average battery life is 9 hours ( ).

Step 4: Analyze the given options in the context of a Type 1 error.
Let’s examine each option:
Option 1: “concluding that the new batteries will make a profit when they actually will not”
This option talks about profit, which is not directly related to the hypothesis test about battery life. The hypothesis test is about the mean battery life, not profit.

Option 2: “concluding that the new batteries last longer than nine hours when they actually do”
This describes a correct decision (Type II error is failing to reject when is true). If we conclude the batteries last longer than 9 hours, and they actually do, this is a correct conclusion.

Option 3: “concluding that the new batteries last longer than nine hours when they actually do not”
This means we conclude when in reality . Since the null hypothesis is and the alternative is , “actually do not” implies that the true mean battery life is not greater than 9 hours, i.e., . If we reject in favor of , and the true mean is actually 9 or less, then we have made a mistake. If the true mean is exactly 9, then rejecting is a Type 1 error. If the true mean is less than 9, we are essentially rejecting when it is false and is also false. However, the context of Type 1 error is specifically about rejecting a true null hypothesis. The most precise definition of a Type 1 error here is concluding when . If the true mean is less than 9, then is false, and concluding would be a Type 1 error if the test was designed to reject when and the actual mean was 9. Let’s refine this. A Type 1 error is rejecting when is true. If is true, then . So, a Type 1 error is concluding when . This option states “when they actually do not”, which implies that the true condition is not . This aligns with a Type 1 error if the true mean is indeed 9.

Option 4: “concluding that the new batteries do not last longer than nine hours when they actually do”
This means we fail to reject (conclude ) when is true ( ). This is a Type 2 error.

Option 5: “concluding that the new batteries do not last longer than nine hours when they actually do not”
This means we fail to reject (conclude ) when is true ( ) or when . If , failing to reject is a correct decision. If , then is false, and failing to reject it is a Type 2 error. This option doesn’t describe a Type 1 error.

Let’s re-evaluate Option 3.
Type 1 error: Reject when is true.


Rejecting means concluding that the batteries last longer than nine hours.
is true means that the true mean battery life is 9 hours.
So, a Type 1 error is concluding that the batteries last longer than nine hours (i.e., ) when the true mean battery life is 9 hours (i.e., ).

Option 3 states: “concluding that the new batteries last longer than nine hours when they actually do not”
If “they actually do not” refers to the claim that “the new batteries last longer than nine hours”, then it means that the conclusion is false. This is consistent with a Type 1 error if the true state is .

Let’s rephrase Option 3 to be more explicit about the null hypothesis being true.
A Type 1 error is concluding that the new batteries last longer than nine hours (rejecting ) when the true average battery life is actually nine hours (i.e., is true).

Let’s check the wording of Option 3 again: “concluding that the new batteries last longer than nine hours when they actually do not”.
This means we conclude , but the true situation is not . The true situation could be or .
If the true mean is , and we conclude , this is a Type 1 error.
If the true mean is , and we conclude , this is also a type of error, but the standard definition of Type 1 error is rejecting a true null hypothesis.

Let’s consider the options in terms of statistical decision and true state.
We are testing vs .
Possible decisions:

Possible true states:

Type 1 error: Reject when is true.
This means concluding when .

Let’s re-examine Option 3: “concluding that the new batteries last longer than nine hours when they actually do not”
This means: Conclude AND the true situation is NOT .
If the true situation is NOT , it means the true mean is .
If the true mean is , and we conclude , this is a Type 1 error.
If the true mean is , and we conclude , this is also an error, but it’s not the definition of a Type 1 error which specifically relates to the null hypothesis being true.

However, in the context of the provided options, Option 3 is the closest to the definition of a Type 1 error. The phrasing “when they actually do not” is slightly ambiguous. If it implies that the true state is , then it’s a perfect fit. If it implies , then it’s also consistent with a Type 1 error.

Let’s consider the alternative hypothesis is that the batteries last longer than nine hours. If we conclude they do, but they actually do not, this is precisely what a Type 1 error is. The null hypothesis is that they last nine hours. The alternative is that they last longer than nine hours. If we conclude they last longer than nine hours, but in reality, they do not (meaning they last nine hours or less), and the null hypothesis was actually true (meaning they last nine hours), then we have made a Type 1 error.

Step 5: Select the best option.
Option 3 is the most appropriate description of a Type 1 error in this scenario. It describes the situation where we conclude that the batteries last longer than nine hours (rejecting the null hypothesis) when, in reality, they do not last longer than nine hours (meaning the null hypothesis is true, or the alternative is false). The most precise definition of a Type 1 error is rejecting a true null hypothesis. Here, the null hypothesis is . Rejecting it means concluding . If the null hypothesis is true, then . So, a Type 1 error is concluding when . Option 3 states “concluding that the new batteries last longer than nine hours when they actually do not”. This means we conclude the batteries last longer than 9 hours, but the actual state is that they do not last longer than 9 hours. This aligns with a Type 1 error.

The final answer is .