Question

use the aleks calculator to solve the following problems.
(a) consider a t distribution with 4 degrees of freedom. compute ( p(-1.56 < t < 1.56) ). round your answer to at least three decimal places.
( p(-1.56 < t < 1.56)=square )
(b) consider a t distribution with 25 degrees of freedom. find the value of ( c ) such that ( p(t leq c)=0.01 ). round your answer to at least three decimal places.
( c=square )

Explanation:

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

Since the t - distribution is symmetric about \(t = 0\), \(P(-1.56\lt t\lt1.56)=1 - 2P(t\geq1.56)\) with \(n = 4\) degrees of freedom. Using the ALEKS calculator (or t - distribution table), for \(n = 4\) and \(t = 1.56\), \(P(t\geq1.56)\approx0.09\). Then \(1-2\times0.09 = 1 - 0.18=0.82\).

Step2: Use the inverse t - distribution function for part (b)

We want to find \(c\) such that \(P(t\leq c)=0.01\) with \(n = 25\) degrees of freedom. Using the ALEKS calculator (inverse t - distribution function), for a one - tailed test with \(\alpha=0.01\) and \(n = 25\) degrees of freedom, we input the values into the calculator.

Answer:

(a) \(P(-1.56\lt t\lt1.56)\approx0.820\)
(b) \(c\approx - 2.485\)