2. Each of the following is a large-sample confidence interval for μ, the mean resonance frequency (HZ) for all tennis rackets of a certain type. (114.4, 115.6), (114.1, 115.9)

a. What is the value of the sample mean resonance frequency?
b. Both intervals were calculated from the same data. The confidence level of one of these intervals is 90% and the other is 99%. Which of these intervals has the 99% confidence level, and why?

To answer these questions, we need to understand what a confidence interval is and how it is calculated.

a. The value of the sample mean resonance frequency can be found by taking the average of the two values provided in each interval. For the first interval (114.4, 115.6), the sample mean can be calculated as:

(114.4 + 115.6) / 2 = 230 / 2 = 115 HZ

Similarly, for the second interval (114.1, 115.9):

(114.1 + 115.9) / 2 = 230 / 2 = 115 HZ

So, the value of the sample mean resonance frequency is 115 HZ for both intervals.

b. The confidence level of a confidence interval represents the level of certainty or probability that the interval contains the true population mean. In this case, we have two intervals with confidence levels of 90% and 99%.

The interval with the wider range, (114.1, 115.9), has a 99% confidence level because it allows for more variability in the data. A wider interval indicates less precision in estimating the population mean, but a higher level of confidence that the estimated interval will contain the true population mean.

On the other hand, the narrower interval, (114.4, 115.6), has a 90% confidence level. A narrower interval indicates more precision in estimating the population mean, but with a lower level of confidence that the interval will contain the true population mean.

Therefore, the interval with the 99% confidence level is (114.1, 115.9) because it allows for a greater margin of error, providing a higher level of confidence that the true population mean is within that range.

a. To find the value of the sample mean resonance frequency, we can take the average of the lower and upper bounds of each confidence interval.

For the first confidence interval (114.4, 115.6), the sample mean resonance frequency is (114.4 + 115.6) / 2 = 230 / 2 = 115 Hz.

For the second confidence interval (114.1, 115.9), the sample mean resonance frequency is (114.1 + 115.9) / 2 = 230 / 2 = 115 Hz.

Therefore, the value of the sample mean resonance frequency is 115 Hz for both intervals.

b. The confidence level of the interval (114.1, 115.9) is 90%, and the confidence level of the interval (114.4, 115.6) is 99%.

The confidence level represents the likelihood that the true population mean falls within the interval. The higher the confidence level, the wider the interval.

In this case, the interval (114.4, 115.6) has the 99% confidence level because it is narrower than the interval (114.1, 115.9). A wider confidence interval indicates a lower confidence level, while a narrower interval indicates a higher confidence level.