Each of the following is a confidence interval for u = true average(population 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 sample data. The confidence level for one of the intervals is 90% and for the other is 99%. Which of the intervals has the 90% confidence level, why?
A) To find the value of the sample mean resonance frequency, you need to calculate the midpoint of each confidence interval. The midpoint is the average of the upper and lower bounds of the interval.
For the first interval (114.4, 115.6):
Midpoint = (114.4 + 115.6) / 2 = 230 / 2 = 115 Hz
For the second interval (114.1, 115.9):
Midpoint = (114.1 + 115.9) / 2 = 230 / 2 = 115 Hz
Therefore, the value of the sample mean resonance frequency for both intervals is 115 Hz.
B) The interval with the 90% confidence level is (114.4, 115.6). In hypothesis testing, the confidence level represents the probability that the interval contains the true population mean. A 90% confidence level means that if we were to repeat the sampling and calculation process many times, we would expect 90% of the resulting intervals to contain the true population mean. The remaining 10% represents the probability of the interval not containing the true mean.
In contrast, the 99% confidence level represents a higher level of confidence, meaning that the probability of the interval containing the true population mean is higher. Therefore, the interval (114.1, 115.9) with the 99% confidence level has a higher level of confidence because it is wider and contains a larger range of values. This wider interval allows for a greater margin of error and increases the chances of containing the true population mean.
So, the interval with the 90% confidence level is (114.4, 115.6) because it has a lower level of confidence compared to the interval (114.1, 115.9) with the 99% confidence level.
A) To find the value of the sample mean resonance frequency, we need to calculate the average of the two intervals.
For the first interval, (114.4, 115.6), the sample mean resonance frequency would be:
(114.4 + 115.6) / 2 = 230 / 2 = 115 Hz
For the second interval, (114.1, 115.9), the sample mean resonance frequency would be:
(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 for an interval represents the level of certainty or reliability that the interval contains the true population mean. In this case, one interval has a confidence level of 90% and the other has a confidence level of 99%.
The interval with a confidence level of 90% is (114.1, 115.9).
This is because the larger the confidence level, the wider the interval needs to be to accommodate for more uncertainty. Since the interval (114.1, 115.9) is wider than (114.4, 115.6), it is likely to have a higher confidence level, which in this case is 99%.
Therefore, the interval (114.1, 115.9) has the 90% confidence level.