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The concept of the sample mean is crucial in statistics, especially when dealing with data analysis. The sample mean is the average value obtained from a set of data points, providing a central measure of the dataset. It's calculated by adding up all the data values and then dividing by the number of data points. In the exercise, the sample mean represents the average resonance frequency of tennis rackets from the provided sample. Here's a simple way to understand it:

• The sample mean is like finding the "middle" of your data distribution.
• For instance, given two endpoints of a confidence interval, we can find the sample mean by averaging these endpoints.

In our case, both intervals provided, (114.4, 115.6) and (114.1, 115.9), yielded the same sample mean of 115 Hz. This tells us the average resonance frequency is consistently observed across different confidence intervals.

Resonance frequency refers to the natural frequency at which a system or object oscillates. For tennis rackets, this means how it vibrates when used. It's a critical factor in their design because it affects performance and feel. Some important points about resonance frequency include:

• Every object has its own specific resonance frequency.
• In manufacturing, maintaining consistent resonance frequencies across products ensures reliability and quality.

In the context of our exercise, understanding the resonance frequency of tennis rackets helps provide insights into how consistent they are in terms of design and performance characteristics. This can impact players' performance and the durability of the rackets.

The confidence level in statistics indicates how certain we are that a parameter lies within a confidence interval. It's fundamental in making predictions and understanding the reliability of statistical estimates. Let's break it down:

• A higher confidence level means more certainty about covering the true parameter but at the cost of a wider interval.
• Conversely, a lower confidence level gives a narrower interval but with less certainty.

In our scenario, the interval (114.1, 115.9) is wider, suggesting it has a 99% confidence level, implying higher certainty. Meanwhile, the interval (114.4, 115.6) is narrower and corresponds to a 90% confidence level, indicating slightly less certainty but a more precise range.