91影视

Hypotheses are typically written in the form of if/then statements, such as if someone consumes a lot of sugar, they will develop cavities in their teeth.

The hypotheses of interest are

\({H_0}\,:\,{\mu _i} = {\mu _j},i \ne j\,\)

versus alternative hypothesis

\({H_a}:\)at least two of the are different,

where \(\,{\mu _1},\,{\mu _2},\,{\mu _3}\) are corresponding true averages modulus of elasticity grade \(1,2,3\)respectively.

Denote with

\(\begin{aligned}{l}{x_i}\sum\limits_{j = 1}^J {{x_{ij}}} \\{x_{..}}\sum\limits_{i = 1}^I {{x_{ij}}} .\end{aligned}\)

The total sum of squares Upgrade

\(\left( {SST} \right),\)

\(\left( {SSTr} \right)\),and

Error sum of squares

\({\rm{ }}\left( {SSE} \right)\) are given by

\(SST = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{{\left( {{x_{ij}} - \overline x ..} \right)}^2}} = } \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{x^2}_{ij} - \frac{1}{{I\,\,.\,J}}} {x^2}_{..;}} \)

\(SSTr = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{{\left( {\overline {{x_{i.}}} - \overline x ..} \right)}^2}} = \frac{1}{J}.} \sum\limits_{j = 1}^J {{x^2}_{i.} - \frac{1}{{I\,\,.\,J}}{x^2}_{..;}} \)

\(SSE = \sum\limits_{i = 1}^I {{{\sum\limits_{j = 1}^J {\left( {{x_{ij}} - \overline {{x_{i.}}} } \right)} }^2}} .\)

The mean squares are

\(\begin{aligned}{l}MSTr = \frac{1}{{I - 1}}.SSTr;\\MSE = \frac{1}{{I.\left( {J - 1} \right)}}.SSE.\end{aligned}\)

F is ratio of the two mean

\(F = \frac{{MSTr}}{{MSE}}.\)

The degree of freedom

\(\begin{aligned}{l}I - 1 = 3 - {\rm{ }}1{\rm{ }} = {\rm{ }}2;\\I.\left( {J - 1} \right) = 3 \cdot \left( {10 - 1} \right) = 27;\\I\,\,.\,J - 1 = 3 \cdot 10 - 1 = 29.\end{aligned}\)

\(\overline X ..\)grand mean is

\(\overline X .. = \frac{1}{{I\,\,.\,\,J}}.\,x.. = \frac{1}{I}\sum\limits_{i = 1}^I {\overline {{x_i}} } {\rm{ }} = \frac{1}{3} \cdot \left( {1.63 + 1.56 + 1.42} \right)\)

\( = 1.3567,\)

The treatment sum of squares is

\(SSTr = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{{\left( {\overline {{x_i}} . - \overline {x..} } \right)}^2}} = J.} {\rm{ }}\sum\limits_{i = 1}^I {{{\left( {\overline {{x_i}} . - \overline x ..} \right)}^2}} \)

\( = 10.{\left( {\left( {1.63 - 1.5367} \right)} \right)^2} + {\left( {1.56 - 1.5367} \right)^2} + {\left. {\left( {1.42 - 1.5367} \right)} \right)^2}\)

\( = 0.2286.\)

Error sum of squares is

\(SSE = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{{\left( {{x_{ij}} - \overline {{x_i}.} } \right)}^2}} = \left( {J - 1} \right).} \left( {{s_1}^2 + {s_2}^2 + {s_3}^3} \right)\)

\(\begin{aligned}{l} = \left( {10 - 1} \right).\left( {{{0.27}^2} + {{0.24}^2} + {{0.26}^2}} \right)\\ = 1.7829.\end{aligned}\)

The mean squares can be computed now as

\(\begin{aligned}{l}MSTr = \frac{1}{{3 - 1}}.SSTr = \frac{1}{2}.0.2286 = 0.1143;\\MSE = \frac{1}{{3.\left( {10 - 1} \right)}}.1.7829 = 0.066.\end{aligned}\)

The value of F Statistic

\(f = \frac{{MSTr}}{{MSE}} = \frac{{0.1143}}{{0.066}} = 1.73.\)

can either make conclusion about the hypotheses look at the F critical value or a P value. Remember that the hypotheses of interest are

\({H_0}\,:\,{\mu _i} = {\mu _j},i \ne j\,\)

versus alternative hypothesis

\({H_a}:\)at least two of the are different

The P value is the area to the right of f value under the curve where F has Fisher's distribution with degrees of freedom \(2\)and\(27\) ;

\(P = P\left( {F > f} \right) = P\left( {F > 1.3} \right) = 0.1964.{\rm{ }}\)

thus which was computed using software (you could estimate it using the table in the appendix). Because \({\rm{P = 0}}{\rm{.1964 > }}\alpha \)

do not reject null hypothesis

at given significance level. There is no statistically significance difference in true averages among three grades.

Hence,

do not reject null hypothesis.