Question

use the aleks calculator to solve the following problems.
(a) consider a t distribution with 11 degrees of freedom. compute ( p(tgeq - 1.22) ). round your answer to at least three decimal places.
( p(tgeq - 1.22)=square )
(b) consider a t distribution with 19 degrees of freedom. find the value of ( c ) such that ( p(-clt tlt c)=0.95 ). round your answer to at least three decimal places.
( c=square )

Explanation:

Step1: Calculate \(P(t\geq - 1.22)\) for \(t\) - distribution with \(df = 11\)

Since the total area under the \(t\) - distribution curve is \(1\), and \(P(t\geq - 1.22)=1 - P(t\lt - 1.22)\). Using the symmetry property of the \(t\) - distribution (\(P(t\lt - a)=P(t\gt a)\) for a symmetric \(t\) - distribution about \(0\)), we can also note that for a \(t\) - distribution with \(n = 11\) degrees of freedom, using the ALEKS calculator (or \(t\) - distribution tables/software), \(P(t\lt - 1.22)\) is the left - tail probability. The right - tail probability \(P(t\geq - 1.22)\) is equivalent to \(1-\) left - tail probability. Using the ALEKS calculator, we find that \(P(t\geq - 1.22)=0.883\)

Step2: Find \(c\) for \(t\) - distribution with \(df = 19\) such that \(P(-c\lt t\lt c)=0.95\)

We know that \(P(-c\lt t\lt c)=1 - 2P(t\gt c)\). If \(P(-c\lt t\lt c)=0.95\), then \(1 - 2P(t\gt c)=0.95\). Solving for \(P(t\gt c)\), we get \(P(t\gt c)=\frac{1 - 0.95}{2}=0.025\). Looking up the \(t\) - value in the \(t\) - distribution table (or using the ALEKS calculator) with \(n = 19\) degrees of freedom and right - tail probability \(0.025\), we find \(c = 2.093\)

Answer:

(a) \(0.883\)
(b) \(2.093\)