Question

(a) consider a t distribution with 28 degrees of freedom. compute ( p(-1.24 < t < 1.24) ). round your answer to at least three decimal places.

( p(-1.24 < t < 1.24) = square )

(b) consider a t distribution with 27 degrees of freedom. find the value of ( c ) such that ( p(t leq c) = 0.05 ). round your answer to at least three decimal places.

( c = square )

Explanation:

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

Since the t - distribution is symmetric about \(t = 0\), \(P(-1.24\lt t\lt1.24)=1 - 2P(t\lt - 1.24)\). Using a t - table or a calculator with a t - distribution function (e.g., in R: 2*pt(1.24,28,lower.tail = TRUE) where pt is the cumulative distribution function of the t - distribution, the degrees of freedom df = 28).

$$P(-1.24\lt t\lt1.24)=1-2\times P(t\lt - 1.24)$$

$$P(-1.24\lt t\lt1.24)=1 - 2\times(1 - P(t\lt1.24))$$

Using a calculator or software, \(P(t\lt1.24,28)\approx0.890\)

$$P(-1.24\lt t\lt1.24)=1-2\times(1 - 0.890)=0.780$$

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

We want to find \(c\) such that \(P(t\leq c)=0.05\) with \(df = 27\). Using a t - table (one - tailed) or a calculator (in R: qt(0.05,27,lower.tail = TRUE) where qt is the quantile function of the t - distribution).
Looking up in a t - table or using software, for a one - tailed test with \(\alpha=0.05\) and \(df = 27\), the value of \(c=-1.703\)

Answer:

(a) \(0.780\)
(b) \(-1.703\)