Question 1165660: For a t-distribution probability density curve with df = 27, find the two t-values that divide the area under the curve into a middle 0.95 area and two outside areas of 0.025 (this is the critical value for a 95% confidence interval based on the t distribution).
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To find the two $t$-values that define the middle $0.95$ area of a $t$-distribution with $df=27$, you are looking for the critical values, $\pm t_{\alpha/2}$.
In this case:
* The area in the middle is **0.95**.
* The total area in the two tails is $1 - 0.95 = 0.05$.
* The area in each individual tail ($\alpha/2$) is $0.05 / 2 = \mathbf{0.025}$.
* The **degrees of freedom ($df$)** are **27**.
You need to find the $t$-value such that the area to the right of it is $0.025$. This value is often denoted as $t_{0.025, 27}$.
Using a $t$-distribution table or calculator for a two-tailed $95\%$ confidence level with $df=27$:
$$\mathbf{t_{\text{critical}} \approx 2.052}$$
Therefore, the two $t$-values that divide the area under the curve are $\mathbf{-2.052}$ and $\mathbf{+2.052}$.
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