T -models, part II Using the t tables, software, or a...

$t$ -models, part II Using the $t$ tables, software, or a calculator, estimate
a) the critical value of $t$ for a $99 \%$ confidence interval with $\mathrm{df}=2$
b) the critical value of $t$ for a $90 \%$ confidence interval with $\mathrm{df}=111$

Key Concepts

t-distribution

The t-distribution is a probability distribution used in statistics when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It has heavier tails than the normal distribution, which accounts for the extra uncertainty especially in small samples.

Degrees of Freedom (df)

Degrees of freedom represent the number of values in a calculation that are free to vary without violating any given constraints. In the context of the t-distribution, the degrees of freedom are usually the sample size minus one, and they play a crucial role in determining the exact shape of the t-distribution and the corresponding critical values.

Confidence Interval

A confidence interval is a range of values that is used to estimate a population parameter, such as a mean, with an associated level of confidence (for example, 90% or 99%). The interval is constructed in such a way that it is expected to contain the true parameter with a specified probability reflecting the confidence level.

Critical Value

The critical value is a threshold derived from the t-distribution (or another relevant distribution) that separates the acceptance region from the rejection region in hypothesis testing or is used to calculate confidence intervals. Its value depends on the chosen confidence level and the degrees of freedom, and it determines the extent of the interval or the decision boundary for statistical tests.

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