Final answer:
The sample mean resonance frequency is the midpoint of the confidence interval, calculated by averaging the lower and upper bounds. A 95 percent confidence interval suggests that 95 percent of such intervals from repeated sampling would contain the true population mean. The width of the interval indicates the estimate's precision.
Step-by-step explanation:
The sample mean resonance frequency is the midpoint of the interval of the confidence interval provided. To find this, we can simply average the lower and upper bounds of the interval.
For instance, for the interval (111.6, 112.4), the sample mean would be (111.6 + 112.4) / 2 = 112 Hz. This is a simple calculation that applies to any such interval regarding confidence levels for population means.
Confidence intervals are a range of values that are used to estimate the true value of a population parameter like the population mean (μ).
When we construct a 95 percent confidence interval, we are saying that if we were to take many samples and construct confidence intervals from each one, 95 percent of those intervals would contain the true population mean. The width of the confidence interval gives us an idea of the precision of our estimate; narrower intervals suggest more precise estimates.
If the question were about a 90 percent confidence interval, the same logic would apply, except that we'd expect 90 percent of the intervals to contain the true population mean. In general, a higher confidence level results in a wider interval because more certainty requires a larger range to ensure the true mean is captured.