Questions: 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.

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.
Transcript text: 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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Solution

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Solution Steps

Step 1: Standard Error Calculation

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{50}} \approx 0.7071 \]

Step 2: Test Statistic Calculation

The test statistic \( Z_{test} \) is calculated using the formula: \[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{32 - 30}{0.7071} \approx 2.8284 \]

Step 3: P-value Calculation

For a right-tailed test, the p-value is calculated as: \[ P = 1 - T(z) \approx 0.0023 \]

Step 4: Interpretation of Results

The results of the hypothesis test are as follows:

  • Test Statistic: \( 2.8284 \)
  • P-value: \( 0.0023 \)
  • Standard Error: \( 0.7071 \)
  • Sample Mean: \( 32 \)
  • Sample Standard Deviation: \( 5 \)
Step 5: Type 2 Error Interpretation

A Type 2 error in this context means that 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.

Final Answer

The answer is: \\(\boxed{\text{A Type 2 error means the local government fails to recognize a true increase in average commute time.}}\\)

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