Determine the t-percentile that is required to construct each of the following two-sided confidence intervals:
(a) Confidence level = 95%, degrees of freedom = 12
(b) Confidence level = 95%, degrees of freedom = 24
(c) Confidence level = 99%, degrees of freedom = 13
(d) Confidence level = 99.9%, degrees of freedom = 15
The Correct Answer and Explanation is :
The t-percentile (critical value for t-distribution) required to construct the two-sided confidence intervals for each of the cases is as follows:
(a) Confidence level = 95%, degrees of freedom = 12:
The t-percentile is approximately 2.179.
(b) Confidence level = 95%, degrees of freedom = 24:
The t-percentile is approximately 2.064.
(c) Confidence level = 99%, degrees of freedom = 13:
The t-percentile is approximately 3.012.
(d) Confidence level = 99.9%, degrees of freedom = 15:
The t-percentile is approximately 4.073.
Explanation:
The t-distribution is used for confidence intervals when the sample size is small and the population standard deviation is unknown. It is characterized by degrees of freedom (df), which typically corresponds to the sample size minus one (n-1). The t-distribution is wider than the normal distribution, accounting for the increased uncertainty in estimating population parameters from a smaller sample.
To construct a two-sided confidence interval, you need the critical t-value, which corresponds to the desired confidence level. This critical value is the point beyond which a certain percentage of the data lies in each tail of the distribution. For a confidence level of 95%, the critical value corresponds to the value where 2.5% of the distribution lies in the upper tail, and 2.5% lies in the lower tail (totaling 5% in both tails). The same reasoning applies for other confidence levels (99%, 99.9%), but with a wider range to account for the higher confidence.
The t-percentile can be found using statistical tables or functions such as scipy.stats.t.ppf, which calculates the percentile (inverse of the cumulative distribution function) for the given confidence level and degrees of freedom. The values obtained are used to calculate the margin of error and construct the confidence interval around the sample mean.