To assess the transportation engineer's claim that the mean commute time is greater than 25 minutes, we will conduct a hypothesis test using the sample data provided. We will use a one-sample z-test for this purpose.
Step 1: Set up the hypotheses
- Null hypothesis \( H_0 \): \( \mu \leq 25 \)
- Alternative hypothesis \( H_a \): \( \mu > 25 \)
Step 2: Given data
- Sample mean (\( \bar{x} \)) = 25.4 minutes
- Population mean (\( \mu_0 \)) = 25 minutes
- Sample standard deviation (\( s \)) = 13 minutes
- Sample size (\( n \)) = 1923
- Significance Level (\( \alpha \)) = 0.01
Step 3: Calculate the test statistic
Since we are dealing with a large sample size (n = 1923), we can use the z-test for the population mean. The test statistic \( z \) is calculated using the formula:
\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]
Where \( \sigma \) is the population standard deviation. Since we only have the sample standard deviation, we will use it here:
\[ z = \frac{25.4 - 25}{13 / \sqrt{1923}} \]
Calculating this step by step:
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Calculate the standard error (SE): \[ SE = \frac{13}{\sqrt{1923}} \approx \frac{13}{43.87} \approx 0.296 \]
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Substitute back into the z formula: \[ z = \frac{25.4 - 25}{0.296} \approx \frac{0.4}{0.296} \approx 1.351 \]
Step 4: Find the P-value
Next, we look up the calculated z-value in the standard normal distribution table or use statistical software.
Using a z-table or calculator:
- The cumulative probability for \( z = 1.351 \) is approximately 0.9115.
- Since we are conducting a one-tailed test, we can find the P-value by: \[ P(Z > 1.351) = 1 - P(Z < 1.351) \approx 1 - 0.9115 \approx 0.0885 \]
Step 5: Make a decision
We'll compare the P-value with the significance level \( \alpha = 0.01 \).
- If \( P \leq \alpha \), we reject \( H_0 \).
- If \( P > \alpha \), we fail to reject \( H_0 \).
In this case:
- P-value \( \approx 0.0885 \) which is greater than \( 0.01 \).
Conclusion
Since the P-value (0.0885) is greater than the significance level (0.01), we do not have enough evidence to reject the null hypothesis. Therefore, we conclude that the data do not provide convincing evidence that the mean commute time is greater than 25 minutes. The engineer's claim is not supported by this analysis.