91Ó°ÊÓ

Upper confidence bound for \({\rm{\mu }}\)and lower confidence bound for \({\rm{\mu }}\)are given by

\(\begin{array}{l}{\rm{\bar x + }}{{\rm{t}}_{{\rm{\alpha ,n - 1}}}}{\rm{ \times }}\frac{{\rm{s}}}{{\sqrt {\rm{n}} }}{\rm{ upper bound, }}\\{\rm{\bar x - }}{{\rm{t}}_{{\rm{\alpha ,n - 1}}}}{\rm{ \times }}\frac{{\rm{s}}}{{\sqrt {\rm{n}} }}{\rm{ lower bound, }}\end{array}\)

with confidence level of \({\rm{100(1 - \alpha )\% }}\)The distribution the random sample is taken from is normal.

(a):

The critical value \({{\rm{t}}_{{\rm{a,n - 1}}}}\)for an upper/lower confidence bound with confidence level\(95\% \), can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 95}}}\\{{\rm{\alpha - 0}}{\rm{.05}}}\end{array}\)

and, from the table A. 5 in the appendix, for \(10\)degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.05}}\)the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.05,10}}}}{\rm{ = 1}}{\rm{.812}}{\rm{.}}\)The area to the left of the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.95}}\).

(b):

The critical value \({{\rm{t}}_{{\rm{\alpha ,n - 1}}}}\)for an upper/lower confidence bound, with confidence level \(95\% \), can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 95}}}\\{{\rm{\alpha - 0}}{\rm{.05}}}\end{array}\)

and, from the table A.5 in the appendix, for 15 degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.05}}\)the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.05,15}}}}{\rm{ = 1}}{\rm{.753}}\)

The area to the left of the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.95}}\).

(c):

The critical value \({{\rm{t}}_{{\rm{\alpha ,n - 1}}}}\) for an upper/lower confidence bound, with confidence level \(99\% \), can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 99}}}\\{{\rm{\alpha - 0}}{\rm{.01}}}\end{array}\)

and, from the table A.5 in the appendix, for \(15\) degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.01}}\) the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.01,15}}}}{\rm{ = 2}}{\rm{.602}}\)

The area to the left of the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.99}}\).

(d):

The critical value \({{\rm{t}}_{{\rm{a,n - 1}}}}\)for an upper/lower confidence bound, with confidence level \(99\% \), can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 99}}}\\{{\rm{\alpha - 0}}{\rm{.01}}}\end{array}\)

and, from the table A. 5 in the appendix, for \({\rm{n = 1 - 4}}\)degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.01}}\) the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.01,4}}}}{\rm{ = 3}}{\rm{.747}}\)

The area to the leftof the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.99}}\)

(e):

The critical value \({{\rm{t}}_{{\rm{a,n - 1}}}}\) for an upper/lower confidence bound, with confidence level \(98\% \), can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 98}}}\\{{\rm{\alpha - 0}}{\rm{.02}}}\end{array}\)

and, from the table A.5 in the appendix, for \(24\) degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.02}}\)$ the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.172,24}}}}{\rm{ = 2}}{\rm{.172}}\)

The area to the left of the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.98}}\).

(f)

The critical value \({{\rm{t}}_{{\rm{a,n - 1}}}}\) for an upper/lower confidence bound, with confidence level \(99\% \)can be obtained as

\(\begin{array}{*{20}{r}}{{\rm{100(1 - \alpha ) - 99}}}\\{{\rm{\alpha - 0}}{\rm{.01}}}\end{array}\)

and, from the table A.5 in the appendix, for \({\rm{38 - 1 = 37}}\)degrees of freedom, and \({\rm{\alpha - 0}}{\rm{.01}}\)the t critical value is

\({{\rm{t}}_{{\rm{0}}{\rm{.01,37}}}}{\rm{ - 2}}{\rm{.429}}\)

The area to the left of the critical value is shaded blue on the picture and has surface area of \({\rm{1 - \alpha - 0}}{\rm{.99}}\).

Hence

\(\begin{array}{l}{\rm{a}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.05,10}}}}{\rm{ - 1}}{\rm{.812;}}\\{\rm{b}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.05,15}}}}{\rm{ - 1}}{\rm{.753; }}\\{\rm{c}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.01,15}}}}{\rm{ - 2}}{\rm{.602; }}\\{\rm{d}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.01,4}}}}{\rm{ - 3}}{\rm{.747;}}\\{\rm{e}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.02,24}}}}{\rm{ - 2}}{\rm{.172; }}\\{\rm{f}}{\rm{. }}{{\rm{t}}_{{\rm{0}}{\rm{.01,37}}}}{\rm{ - 2}}{\rm{.429}}\end{array}\)