Question

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

Explanation:

Step1: Use the property of t - distribution for part (a)

For a t - distribution, \(P(t\geq - 1.22)=1 - P(t\lt - 1.22)\). Using the ALEKS calculator (or t - distribution table/software), with \(n = 11\) degrees of freedom, we find \(P(t\lt - 1.22)\). Then \(P(t\geq - 1.22)=1-0.120 = 0.880\) (using the calculator: input the degree of freedom \(df = 11\), and the value \(x=-1.22\) for the left - tailed probability, then subtract from 1).

Step2: Use the property of symmetric t - distribution for part (b)

Since the t - distribution is symmetric about \(t = 0\), \(P(-c\lt t\lt c)=0.95\) implies that the area in the two tails is \(1 - 0.95=0.05\). So the area in each tail is \(\frac{1 - 0.95}{2}=0.025\). We need to find the \(t\) - value \(c\) such that \(P(t\gt c)=0.025\) with \(df = 19\) degrees of freedom. Using the ALEKS calculator (or t - distribution table/software), input \(df = 19\) and the right - tailed probability \(0.025\), we get \(c = 2.093\).

Answer:

(a) \(P(t\geq - 1.22)=0.880\)
(b) \(c = 2.093\)