Question

a commonly cited standard for one - way length (duration) of school bus rides for elementary school children is 30 minutes.
a local government office in a rural area conducts a study to determine if elementary schoolers in their district have a longer average one - way commute time. if they determine that the average commute time of students in their district is significantly higher than the commonly cited standard they will invest in increasing the number of school busses to help shorten commute time. what would a type 2 error mean in this context?
the local government decides that the average commute time is 30 minutes.
the local government decides that the data provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact 30 minutes.
the local government decides that the data do not provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact higher than 30 minutes.
the local government decides that the data do not provide convincing evidence of an average commute time different than 30 minutes, when the true average commute time is in fact 30 minutes.
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Explanation:

A Type 2 error occurs when we fail to reject a false null hypothesis. In hypothesis testing, the null hypothesis \(H_0\) would be that the average commute time \(\mu = 30\) minutes (the commonly - cited standard), and the alternative hypothesis \(H_1\) would be that \(\mu>30\) minutes (the local government's claim).

• Option 1: Deciding the average commute time is 30 minutes is just stating the null hypothesis, not an error.
• Option 2: This is a Type 1 error (rejecting a true null hypothesis).
• Option 3: Failing to reject the null hypothesis (that the average commute time is 30 minutes) when the null hypothesis is false (the true average is higher than 30 minutes) is a Type 2 error.
• Option 4: Failing to reject a true null hypothesis is not an error.

Answer:

The local government decides that the data do not provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact higher than 30 minutes.