Question

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

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

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.

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%.