Final answer:
To construct 90%, 95%, and 99% confidence intervals using a Student's t-distribution, one needs the sample mean, sample standard deviation, sample size, and the corresponding t-values for the desired confidence levels. The intervals widen with higher confidence levels, and the error bound is the difference between the sample mean and the interval's endpoints.
Step-by-step explanation:
When the population standard deviation (\(σ\)) is unknown and the sample size (\(n\)) is 30 or greater, it's appropriate to use a Student's t-distribution for constructing confidence intervals for the population mean (\(μ\)). To construct a confidence interval for a certain confidence level, you need the sample mean (\(\bar{x}\)), the sample standard deviation (\(s\)), the sample size (\(n\)), and the t-score corresponding to the desired level of confidence and degrees of freedom (\(df\)) which is \(n - 1\).
The formula for a confidence interval using the Student's t-distribution is:
CI = \(\bar{x}\) ± (t-score * (\(s/\sqrt{n}\)))
For example, to construct a 95 percent confidence interval, find the appropriate t-value that corresponds to a two-tailed test with 95 percent of the probability in the middle (leaving 2.5 percent in each tail). Then apply it to the formula above. If the sample mean is 41, the confidence interval would be calculated with the found t-value. The 90 percent, 95 percent, and 99 percent confidence intervals are wider as the confidence level increases because they encompass more of the probability distribution to ensure a higher confidence that the interval contains the true population mean.
For instance, if a previous study's 90 percent confidence interval is (67.18, 68.82) and the sample mean is 68, the error bound is the difference between the sample mean and the endpoints of the interval.