This is a hypothesis testing problem comparing the means of two independent samples (drivers in the large city and rural area) under the assumption that the population variances are equal. This setup calls for a two-sample t-test.

Step 1: State the hypotheses

• Null Hypothesis (H₀): The mean insurance rates in the large city and the rural area are equal. In mathematical terms, $H_0: \mu_{city} = \mu_{rural}$.
• Alternative Hypothesis (H₁): The mean insurance rate in the large city is higher than in the rural area. In mathematical terms, $H_1: \mu_{city} > \mu_{rural}$.

This is a one-tailed test since the claim is directional (higher rates in the city).

Step 2: Calculate the test statistic

The formula for the test statistic for a two-sample t-test, assuming equal variances, is:

$t = \frac{(\bar{X}_{city} - \bar{X}_{rural})}{\sqrt{s_p^2 \left(\frac{1}{n_{city}} + \frac{1}{n_{rural}}\right)}}$

Where:

• $\bar{X}_{city} = 1832$ (mean rate for city drivers)
• $\bar{X}_{rural} = 1609$ (mean rate for rural drivers)
• $s_{city} = 352$ (standard deviation for city drivers)
• $s_{rural} = 232$ (standard deviation for rural drivers)
• $n_{city} = 27$ (number of drivers in the city)
• $n_{rural} = 25$ (number of drivers in the rural area)

The pooled standard deviation $s_p$ is calculated using:

$s_p^2 = \frac{(n_{city} - 1)s_{city}^2 + (n_{rural} - 1)s_{rural}^2}{(n_{city} - 1) + (n_{rural} - 1)}$

Let’s compute these values to find the test statistic.The calculated test statistic is $t = 2.674$ (rounded to three decimal places).

Step 3: Determine the critical value

Since this is a one-tailed test at $\alpha = 0.05$, we need to find the critical value from the t-distribution table for a one-tailed test. The degrees of freedom ($df$) for this test is given by:

$df = (n_{city} - 1) + (n_{rural} - 1) = 27 - 1 + 25 - 1 = 50$

Using $df = 50$, we will look up the critical value for a one-tailed t-test at $\alpha = 0.05$. The critical value is approximately $t_{\text{critical}} = 1.676$.

Step 4: Make a decision

• If $t_{\text{statistic}} > t_{\text{critical}}$, we reject the null hypothesis.
• If $t_{\text{statistic}} \leq t_{\text{critical}}$, we fail to reject the null hypothesis.

Since $t = 2.674$ is greater than $1.676$, we reject the null hypothesis.

Conclusion:

There is sufficient evidence at the $\alpha = 0.05$ level to conclude that drivers in the large city pay higher insurance rates than those in the rural area.

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Would you like more details on any part of this? Here are five related questions:

1. What happens if the population variances are unequal?
2. How is the pooled variance calculated in detail?
3. What is the difference between one-tailed and two-tailed tests?
4. How would the test change if we used a different significance level (e.g., $\alpha = 0.01$)?
5. Can this test be applied if the sample sizes were vastly different?

Tip: Always ensure that the assumption of equal variances is checked before using a pooled t-test.