91Ó°ÊÓ

Given:

\(\begin{array}{l}{\rm{n = 26}}\\{\rm{\bar x = 370}}{\rm{.69}}\\{\rm{s = 24}}{\rm{.36}}\\{\rm{c = 95\% = 0}}{\rm{.95}}\end{array}\)

Determine the t-value by looking in the row starting with degrees of freedom \({\rm{df = n - 1 = 26 - 1 = 25}}\)and in the column with \({\rm{\alpha = 1 - c = 0}}{\rm{.05}}\) in the table of the critical values for t distributions in the appendix:

\({{\rm{t}}_{\rm{\alpha }}}{\rm{ = 1}}{\rm{.708}}\)

The margin of error is then:

\({\rm{E = }}{{\rm{t}}_{\rm{\alpha }}}{\rm{ \times }}\frac{{\rm{s}}}{{\sqrt {\rm{n}} }}{\rm{ = 1}}{\rm{.708 \times }}\frac{{{\rm{24}}{\rm{.36}}}}{{\sqrt {{\rm{26}}} }}{\rm{\gg 8}}{\rm{.1598}}\)

Hence The upper boundary of the confidence interval then becomes:

\({\rm{\bar x + E = 370}}{\rm{.69 + 8}}{\rm{.1598 = 378}}{\rm{.8498}}\)

Given:

\(\begin{array}{l}{\rm{n = 26}}\\{\rm{\bar x = 370}}{\rm{.69}}\\{\rm{s = 24}}{\rm{.36}}\\{\rm{c = 95\% = 0}}{\rm{.95}}\end{array}\)

Upper confidence bound found in part (a): \(378.8498\)

UPPER PREDICTION BOUND

Determine the t-value by looking in the row starting with degrees of freedom \({\rm{df = n - 1 = 26 - 1 = 25}}\) and in the column with \({\rm{\alpha = 1 - c = 0}}{\rm{.05}}\)in the table of the critical values for t distributions in the appendix:

\({{\rm{t}}_{{\rm{\alpha /2}}}}{\rm{ = 1}}{\rm{.708}}\)

The margin of error is then:

\({\rm{E = }}{{\rm{t}}_{{\rm{\alpha /2}}}}{\rm{ \times s}}\sqrt {{\rm{1 + }}\frac{{\rm{1}}}{{\rm{n}}}} {\rm{ = 1}}{\rm{.708 \times 24}}{\rm{.36}}\sqrt {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{26}}}}} {\rm{\gg 42}}{\rm{.3995}}\)

Hence The upper boundary of the prediction interval then becomes:

\({\rm{\bar x + E = 370}}{\rm{.69 + 42}}{\rm{.3995 = 413}}{\rm{.0895}}\)

Thus we note that the upper prediction bound is higher than the upper confidence bound.

(c):

The new random variable is

\({{\rm{\bar X}}_{{\rm{new }}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\left( {{{{\rm{\bar X}}}_{{\rm{27}}}}{\rm{ + }}{{{\rm{\bar X}}}_{{\rm{28}}}}} \right)\)

In order to obtain PI for a single value, standard deviation of random variable

\({\rm{\bar X - }}{{\rm{\bar X}}_{{\rm{new }}}}\)

is needed. When the variance is computed, the t statistic can be obtained.

The variance is

\(\begin{array}{l}{\rm{V}}\left( {{\rm{\bar X - }}{{{\rm{\bar X}}}_{{\rm{new }}}}} \right){\rm{ }}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{V(\bar X) + ( - 1}}{{\rm{)}}^{\rm{2}}}{\rm{V}}\left( {{{{\rm{\bar X}}}_{{\rm{new }}}}} \right){\rm{ = V(\bar X) + V}}\left( {\frac{{\rm{1}}}{{\rm{2}}}\left( {{{{\rm{\bar X}}}_{{\rm{27}}}}{\rm{ + }}{{{\rm{\bar X}}}_{{\rm{28}}}}} \right)} \right)\\\mathop {\rm{ = }}\limits^{{\rm{(2)}}} {\rm{V(\bar X) + }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{V}}\left( {{{{\rm{\bar X}}}_{{\rm{27}}}}} \right){\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{V}}\left( {{{{\rm{\bar X}}}_{{\rm{28}}}}} \right)\\\mathop {\rm{ = }}\limits^{{\rm{(3)}}} \frac{{\rm{1}}}{{\rm{n}}}{{\rm{\sigma }}^{\rm{2}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{\rm{\sigma }}^{\rm{2}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{\rm{\sigma }}^{\rm{2}}}{\rm{ = }}{{\rm{\sigma }}^{\rm{2}}}\left( {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} \right)\end{array}\)

(1): random variables are independent,

(2): random variables are independent and \({\rm{V(cX) = }}{{\rm{c}}^{\rm{2}}}{\rm{V(X)}}\),

(3) : random variables are from the same distribution with variance \({{\rm{\sigma }}^{\rm{2}}}\)and the result for variance of \({\rm{\bar X}}\)is known.

This indicates that the random variable

\({\rm{T = }}\frac{{{\rm{\bar X - }}{{{\rm{\bar X}}}_{{\rm{new }}}}}}{{\sqrt {{{\rm{s}}^{\rm{2}}}\left( {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} \right)} }}\)

has student's t distribution with \({\rm{n - 1}}\)degrees of freedom, where the sample standard deviation is an estimate of the standard deviation.

Therefore, the two sided prediction interval is

\(\left( {{\rm{\bar x - }}{{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ \times s \times }}\sqrt {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} {\rm{,\bar x + }}{{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ \times s \times }}\sqrt {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} } \right)\)

The values are known from the exercise are

\(\begin{array}{l}{\rm{x = 370}}{\rm{.69}}\\{\rm{s = 24}}{\rm{.36,n = 26}}\end{array}\)

The value of \({{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ = }}{{\rm{t}}_{{\rm{0}}{\rm{.025,25}}}}{\rm{ = 2}}{\rm{.06}}\)can be found in the appendix of the book or computed by PC, where \({\rm{\alpha /2 = 0}}{\rm{.025}}\), because the \({\rm{95\% }}\)two-sided interval needs to be computed and

\(\begin{array}{l}{\rm{100(1 - \alpha ) = 95 }}\\{\rm{\alpha = 0}}{\rm{.05}}\end{array}\)

and \({\rm{n - 1 = 26 - 1 = 25}}\)degrees of freedom.

The confidence interval becomes

\(\begin{array}{l}\left( {{\rm{\bar x - }}{{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ \times s \times }}\sqrt {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} {\rm{,\bar x + }}{{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ \times s \times }}\sqrt {\frac{{\rm{1}}}{{\rm{n}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} } \right)\\{\rm{ = }}\left( {{\rm{370}}{\rm{.69 - 2}}{\rm{.06 \times 24}}{\rm{.36 \times }}\sqrt {\frac{{\rm{1}}}{{{\rm{26}}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} {\rm{,370}}{\rm{.69 + 2}}{\rm{.06 \times 24}}{\rm{.36 \times }}\sqrt {\frac{{\rm{1}}}{{{\rm{26}}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}} } \right)\\{\rm{ = (370}}{\rm{.69 - 36}}{\rm{.82,370}}{\rm{.69 + 36}}{\rm{.82)}}\\{\rm{ = (333}}{\rm{.87,407}}{\rm{.51)}}\end{array}\)

Hence The confidence interval becomes\((333.87,407.51)\)