State the total degrees of freedom for the following t...

State the total degrees of freedom for the following $t$ tests: a. $n=12$ for a one-sample $t$ test b. $d f_{1}=12, n_{2}=19$ for a two-independent-sample $t$ test c. Critical value $=63.657$ for a two-tailed test, $\mathrm{a}=.01$ d. Critical value $=1.645$ for a one-tailed test, $\mathrm{a}=.05$

State the total degrees of freedom for the following $t$ tests:
a. $n=12$ for a one-sample $t$ test
b. $d f_{1}=12, n_{2}=19$ for a two-independent-sample $t$ test
c. Critical value $=63.657$ for a two-tailed test, $\mathrm{a}=.01$
d. Critical value $=1.645$ for a one-tailed test, $\mathrm{a}=.05$

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Key Concepts

Degrees of Freedom

Degrees of freedom represent the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the context of t tests, it typically depends on the sample size(s) and affects the shape of the t distribution used to determine critical values and p-values.

One?Sample t Test

A one?sample t test compares the mean of a single sample to a known or hypothesized population mean. Its degrees of freedom are calculated as the sample size minus one, reflecting the number of values that are free to vary when estimating the sample standard deviation.

Two?Independent Sample t Test

A two?independent sample t test is used to compare the means of two independent groups. In most cases, the total degrees of freedom are determined by adding the sample sizes of both groups together and then subtracting 2, one for each estimated sample mean in the pooled variance calculation.

Critical t Value and Significance Level

The critical t value is the threshold value from the t distribution, which, based on the chosen level of significance (alpha), determines the rejection region for a hypothesis test. It is selected according to whether the test is one?tailed or two?tailed, and it depends on the degrees of freedom.

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