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Chapter 12: Problem 21

The two intervals (114.4,115.6) and (114.1,115.9) are confidence intervals for \(\mu=\) mean resonance frequency (in hertz) for all tennis rackets of a certain type. The two intervals were calculated using the same sample data. a. What is the value of the sample mean resonance frequency? b. The confidence level for one of these intervals is \(90 \%,\) and for the other it is \(99 \%\). Which is which, and how can you tell?

Short Answer

Expert verified
a. The sample mean resonance frequency is 115 Hz. b. The interval (114.1, 115.9) corresponds to a confidence level of 99%, and the interval (114.4, 115.6) corresponds to a confidence level of 90%.

Step by step solution

01

Finding the Sample Mean

The sample mean is essentially the average of the two endpoints of the interval. Since the same sample data was used for both intervals, the mean will be the same for both. We can find the average by adding the two endpoints and then dividing by 2. For the first interval, this is \(((114.4 + 115.6) / 2 = 115)\). For the second interval, this is \(((114.1 + 115.9) / 2 = 115)\). Thus, the sample mean resonance frequency is 115 Hz.
02

Identify the Confidence Levels

For the confidence levels, we can safely say that the larger interval corresponds to the higher confidence level. This is because a higher confidence level means more certainty, which comes at the cost of a larger range of potential values. So, the interval (114.1, 115.9) corresponds to a confidence level of 99%, while the interval (114.4, 115.6) corresponds to a confidence level of 90%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean
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
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.
Confidence Level
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.

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Most popular questions from this chapter

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