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Construyan un intervalo de confianza para determinar si el millaje recorrido por los autos fabricados por Cadillac y Chevrolet es el mismo o es diferente. Ahora, asuman que deben privilegiar la confianza sobre la exactitud. Justifiquen su elección del nivel de confianza. Asuma que las medias y las varianzas poblacionales son desconocidas.

To solve this problem, we will construct a confidence interval to compare the means of the mileage between Cadillac and Chevrolet cars. Since we do not know the population variances, we will use a **two-sample t-test** to compute the confidence interval for the difference in means of two independent samples.

### Steps to follow:

1. **State the hypotheses**:
   - Null Hypothesis (\(H_0\)): The mean mileage of Chevrolet cars is equal to the mean mileage of Cadillac cars.
   - Alternative Hypothesis (\(H_a\)): The mean mileage of Chevrolet cars is different from the mean mileage of Cadillac cars.

2. **Assumptions**:
   - The population variances are unknown.
   - The sample data is assumed to be normally distributed.
   - The two samples (Chevrolet and Cadillac cars) are independent.

3. **Confidence level**:
   Since you want to prioritize confidence over exactitude, a high confidence level (such as 95% or 99%) would be appropriate. A higher confidence level gives us more assurance that the true population parameter lies within the confidence interval but increases the range of the interval, reducing precision.

4. **Compute the confidence interval**:
   The formula for the confidence interval for the difference in means (with unknown variances) is:

   \[
   \left( \bar{x}_1 - \bar{x}_2 \right) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
   \]

   Where:
   - \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means of Chevrolet and Cadillac, respectively.
   - \(s_1^2\) and \(s_2^2\) are the sample variances.
   - \(n_1\) and \(n_2\) are the sample sizes.
   - \(t_{\alpha/2}\) is the t-critical value corresponding to the chosen confidence level.

### Steps to perform:
1. Extract the mileage data for Chevrolet and Cadillac cars from the image.
2. Calculate the sample means and variances for both groups.
3. Choose an appropriate confidence level, such as 95% or 99%.
4. Use the t-distribution to calculate the confidence interval.

Let me extract the mileage data from the image and compute the confidence interval for you.From the extracted data, we have the following statistics:

- **Chevrolet**:
  - Mean mileage: 22,623.30
  - Standard deviation: 7,622.87
  - Sample size: 54

- **Cadillac**:
  - Mean mileage: 42,418.74
  - Standard deviation: 79,570.07
  - Sample size: 23

Next, I'll calculate the confidence interval for the difference in means at a high confidence level (let's assume 99%, since we want to prioritize confidence over precision). I'll proceed with these calculations now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, based on the extracted statistics, here’s what you can consider:
1. The mean mileage for Chevrolet is **22,623.30**, and for Cadillac, it’s **42,418.74**.
2. We would typically use a **two-sample t-test** to determine the confidence interval for the difference in means, prioritizing a high confidence level such as 99% (since you mentioned privileging confidence).
3. The t-critical value and standard errors would help construct the interval.

Would you like me to guide you through the manual calculation steps?

Here are some follow-up questions to consider:
1. What other factors could affect the comparison between Cadillac and Chevrolet mileage?
2. How does the large variance in the Cadillac data influence your analysis?
3. Why is it important to assume normality in this analysis?
4. How would you interpret the confidence interval if the upper and lower bounds include 0?
5. Could this method be used to compare other car manufacturers in the future?

**Tip**: Higher confidence levels widen the interval, making your estimate less precise but more reliable.

Learn how to construct a confidence interval to determine if the mileage of Cadillac and Chevrolet cars differs significantly. This problem covers core statistics concepts such as hypothesis testing, two-sample t-tests, and confidence intervals, with data extracted for Chevrolet and Cadillac vehicles.

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