(a) Find the t -value such that the area in the right tail...

(a) Find the $t$ -value such that the area in the right tail is 0.02 with 19 degrees of freedom. (b) Find the $t$ -value such that the area in the right tail is 0.10 with 32 degrees of freedom. (c) Find the $t$ -value such that the area left of the $t$ -value is 0.05 with 6 degrees of freedom. [Hint: Use symmetry.] (d) Find the critical $t$ -value that corresponds to $95 \%$ confidence. Assume 16 degrees of freedom.

(a) Find the $t$ -value such that the area in the right tail is 0.02 with 19 degrees of freedom.
(b) Find the $t$ -value such that the area in the right tail is 0.10 with 32 degrees of freedom.
(c) Find the $t$ -value such that the area left of the $t$ -value is 0.05 with 6 degrees of freedom. [Hint: Use symmetry.]
(d) Find the critical $t$ -value that corresponds to $95 \%$ confidence. Assume 16 degrees of freedom.

Key Concepts

Critical Value

A critical value is a threshold in a probability distribution beyond which the probability of occurrence is low. In the context of the t-distribution, critical values are determined based on the desired level of significance or confidence level, often using symmetry for two-tailed tests, and are crucial for making decisions in hypothesis tests and constructing confidence intervals.

Tail Probability

Tail probability refers to the likelihood of a statistic falling within a specific tail (either right or left) of a distribution. It is essential in hypothesis testing and determining critical values since it helps set the boundaries for rejection regions in statistical tests.

Student’s t-distribution

The t-distribution is a probability distribution used 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 is characterized by its heavier tails compared to the normal distribution, which reflects the added uncertainty from using sample data.

Degrees of Freedom

Degrees of freedom, usually calculated as the sample size minus one, determine the shape of the t-distribution. They reflect the number of independent pieces of information used to estimate a parameter and affect the thickness of the tails; fewer degrees of freedom result in a distribution with heavier tails.

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