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When comparing two different models of cars using confidence intervals, one might wonder which model's interval comes from a larger sample size.
In the context of statistics, the width of a confidence interval is key to understanding sample size.
Given two confidence intervals of these car models, here's what we know:

• The narrower the interval, the larger the sample size it represents.
• Conversely, a wider interval suggests a smaller sample size.

In our exercise, Model 1 has a confidence interval width of 1.4, and Model 2 has a width of 1.2.
Now considering that a smaller width indicates a larger sample, Model 2's narrower interval (49.4, 50.6) implies it was based on a larger sample size.
This illustrates how comparing interval widths helps infer which sample size is larger, enhancing the reliability of the associated mean estimate.

Statistical inference is a cornerstone of understanding and working with confidence intervals.
It involves making predictions or decisions about a population based on sample data.
This method allows researchers to make educated guesses regarding population parameters through statistical models.

• Confidence intervals provide a range within which the true population parameter is estimated to lie.
• The interval is calculated based on the sample data, incorporating the mean and the variability of the data.

In simpler terms, when interpreting a confidence interval, we say that if we were to take multiple samples and compute confidence intervals, approximately 95% of them would contain the true mean.
This interpretation assumes a standard level of confidence, commonly 95%, as seen with our car models.
Understanding this process is essential for predicting and verifying data-driven conclusions without examining an entire population.

Understanding the width of confidence intervals is crucial for analyzing statistical data.
Confidence interval width has an intrinsic connection to sample size and variability:

• The width is influenced by the standard deviation and sample size, given constant confidence level.
• A smaller standard deviation or larger sample size results in a narrower interval.

Mathematically, the width can be expressed as:\[ \text{Width} = 2 \times z^* \times \frac{s}{\sqrt{n}} \]where:

• \(z^*\) is the z-value corresponding to the confidence level (e.g., 1.96 for 95%).
• \(s\) is the standard deviation of the sample.
• \(n\) is the sample size.

The formula reveals that increasing the sample size \(n\) without changing other factors will shrink the confidence interval, suggesting more precise estimates of the population mean.
This is why a narrower confidence interval, like that of Model 2, indicates a larger sample size in our problem.
Understanding this relationship allows statisticians and researchers to effectively interpret and utilize interval data.