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This problem involves hypothesis testing, specifically a **one-sample t-test** to verify if the observed mean reduction in delivery time is statistically significantly less than the claimed 10 minutes. Let's walk through the steps required to solve it.

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### Step 1: Define the Hypotheses
- **Null hypothesis** \(H_0\): The mean reduction in delivery time is at least 10 minutes.  
  \[ H_0: \mu \geq 10 \]
- **Alternative hypothesis** \(H_1\): The mean reduction in delivery time is less than 10 minutes.  
  \[ H_1: \mu < 10 \]

This is a **left-tailed test** since the researcher is interested in determining if the reduction is significantly less than 10 minutes.

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### Step 2: Given Information
- Sample size: \(n = 100\)
- Sample mean: \(\bar{x} = 9.5\)
- Claimed population mean: \(\mu_0 = 10\)
- Sample standard deviation: \(s = 3\)
- Significance level: \(\alpha = 0.05\)

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### Step 3: Compute the Test Statistic
We will use the **t-statistic** formula for a one-sample test:

\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Substitute the given values:

\[
t = \frac{9.5 - 10}{\frac{3}{\sqrt{100}}} = \frac{-0.5}{0.3} = -1.67
\]

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### Step 4: Determine the Critical Value
Since this is a left-tailed test with \( \alpha = 0.05 \) and degrees of freedom \(df = n - 1 = 99\), we need the critical value for \(t_{0.05, 99}\). 

Using a t-distribution table or a calculator:
\[
t_{0.05, 99} \approx -1.660
\]

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### Step 5: Decision Rule
- If the calculated \(t\)-statistic is less than the critical value, we **reject** the null hypothesis.
- If the calculated \(t\)-statistic is greater than the critical value, we **fail to reject** the null hypothesis.

In this case:
\[
t = -1.67 \quad \text{and} \quad t_{critical} = -1.660
\]

Since \(-1.67 < -1.660\), we **reject the null hypothesis**.

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### Step 6: Conclusion
At the 0.05 significance level, there is enough evidence to conclude that the mean reduction in delivery time is **less than** the claimed 10 minutes.

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Do you have any questions or would you like further clarifications on any part?

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Here are five follow-up questions to expand your understanding:
1. How would the result change if the sample size was smaller?
2. What is the p-value for this test, and how does it relate to the conclusion?
3. How would a two-tailed test differ from this one-tailed test?
4. Why did we use the t-distribution instead of the normal distribution?
5. What are Type I and Type II errors in this context?

**Tip:** For larger samples (e.g., \(n > 30\)), the t-distribution closely approximates the normal distribution, which makes t-tests very flexible!

A logistics company claims that a newly implemented routing optimization system reduces delivery times by an average of 10 minutes. To verify this claim, an independent logistics researcher conducted a study on a sample of 100 delivery routes after the system was implemented. The sample showed a mean reduction in delivery time of 9.5 minutes, with a standard deviation of 3 minutes. The researcher wishes to test the hypothesis that the mean reduction in delivery time is less than the claimed 10 minutes, using a 0.05 significance level.

Test the claim that a routing optimization system reduces delivery times by 10 minutes. Learn how to perform a one-sample t-test with a sample mean of 9.5 minutes and a standard deviation of 3 minutes. This problem demonstrates hypothesis testing with a 0.05 significance level and explores key statistical concepts.

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