Final answer:
The alternating series estimation theorem can be used to find an upper bound for the error in series approximations by using the term immediately after the last term used in the approximation as an upper bound. For confidence intervals, the sample mean and error bound are derived from the endpoints of the interval.
Step-by-step explanation:
When working with series approximations such as the alternating series, the alternating series estimation theorem provides a way to find an upper bound for the error in approximations. To apply this theorem, you look at the absolute value of the first term not included in the approximation. This term serves as the upper bound for the approximation error.
Estimating Sample Mean and Error Bound using Confidence Intervals
To estimate the sample mean using a confidence interval, you can subtract the error bound from the upper value of the confidence interval. Alternatively, you can average the upper and lower endpoints of the interval. When finding the error bound, you can subtract the sample mean from the upper value of the interval, or calculate the average difference between the upper and lower values, and then divide by 2.
In the event that the sample size changes, the error bound will also change accordingly. Considering a larger sample size generally results in a smaller error bound, providing a more accurate approximation.
For example, given a confidence interval of (42.12, 47.88), the sample mean would be the average of these two values, and the error bound would be half the difference between them.