Question

a t distribution with 6 degrees of freedom is graphed below. the region under the curve to the right of t = 2 is shaded. find the area of the shaded region. round your answer to three decimal places.

Explanation:

Step1: Use t - distribution table or calculator

We can use a t - distribution table or a statistical calculator. For a t - distribution with \(n = 6\) degrees of freedom, we want to find \(P(T>2)\) where \(T\) follows a \(t\) - distribution with \(df=6\).
Using a calculator (e.g., in R: 1 - pt(2,6)), or a t - table (which gives the cumulative probability \(P(T\leq t)\)).

Step2: Calculate the probability

The cumulative distribution function \(F(t)\) for a \(t\) - distribution gives \(P(T\leq t)\). We know that \(P(T > 2)=1 - P(T\leq2)\).
For \(df = 6\) and \(t = 2\), using a calculator:
\(P(T\leq2)\approx0.920\) (from calculator or more accurate table values). Then \(P(T > 2)=1 - 0.920=0.080\)

Answer:

\(0.080\)