equation-012
equation-012
\( y = a + bx \)
\( x = c + dy \)
\( y = a_{1}x_{1} + a_{2}x_{2} + \cdots +a_{n}x_{n} + b \)
\( y = a + bx + cx^2 \)
\( ( x_{ i } – \bar{ x } )^2 \)
\( ( y_{ i } – \bar{ y } )^2 \)
\( ( x_{ i } – \bar{ x } )( y_{ i } – \bar{ y } ) \)
\( b = \displaystyle\frac{ S_{(xy)} }{ S_{(xx)} } = \displaystyle\frac{ \sum_{i=1}^{n} ( x_{ i } – \bar{ x } )( y_{ i } – \bar{ y } ) }{ \sum_{i=1}^{n} ( x_{ i } – \bar{ x } )^2 } \)
\( a = \bar{ y } – b \bar{ x } \)
\( \bar{x} と \bar{y} の計算 \)
\( \bar{x} \) \( \bar{y} \)
\( \bar{x} = 35.93 \) \( \bar{y} = 14.05 \)
\( S_{(xx)} と S_{(xy)} の計算 \)
\( S_{(xx)} = 231.907 \) \( S_{(xy)} = -197.122 \) \( S_{(yy)} = 177.067 \)
\( b = \displaystyle\frac{ S_{(xy)} }{ S_{(xx)} } = \displaystyle\frac{ -197.122 }{ 231.907 } = -0.850 \)
\( a = \bar{y} – b \bar{x} = 14.05 – (-0.850) \times 35.93 = 44.597 \)
\( y = 44.597 – 0.850 x \)
\( y = \beta_{0} + \beta_{1} x + \varepsilon \)
\( \varepsilon \)
\( E( \varepsilon ) = 0 \)
\( V( \varepsilon ) = \sigma_{ \varepsilon }^2 \)
\( y = a + bx \)
\( x \) \( y \) \( x_{i} \) \( y_{i} \)
\( y_{i} = a + bx_{i} \)
\( ( x_{i}, y_{i} ) \) \( ( a + bx_{i} ) \) \( y_{i} – ( a + bx_{i} ) \)
\( 残差 = y_{i} – ( a + bx_{i} ) \)
\( Q = 残差^2 の合計 = \sum_{i=1}^{n} e_{i}^2 = \sum_{i=1}^{n} | y_{i} – ( a + bx_{i})|^2 \)
\( x = c + dy \)
\( y = – \displaystyle\frac{c}{d} + \displaystyle\frac{1}{d}x \)
\( y = 44.597 – 0.850 x \)
\( d = \displaystyle\frac{ S_{(xy)} }{ S_{(yy)} } = \displaystyle\frac{ -197.122 }{ 177.067 } = -1.113 \)
\( c = \bar{x} – b \bar{y} = 35.93 – (-1.113) \times 14.05 = 51.578 \)
\( x = 51.578 – 1.113 x \)
\( y = 46.331 – 0.898 x \)
\( 寄与率 = (相関係数)^2 = r^2 \)
\( 寄与率 = \displaystyle\frac{ S_{ (xy) }^2 }{ S_{ (xx) }S_{ (yy) }} \)
\( 寄与率 = \displaystyle\frac{ S_{ (xy) }^2 }{ S_{ (xx)}S_{ (yy) }} = \displaystyle\frac{ (-197.122)^2 }{ 231.907 \times 177.067 } = 0.9463 \)
\( S_{R} = \displaystyle\frac{ S_{ (xy) }^2 }{ S_{ (xx) }} = \displaystyle\frac{ (-197.122 )^2 }{ 231.907 } = 167.555 \)
\( S_{T} = S_{ (yy) } = 177.067 \)
\( S_{E} = S_{T} – S_{R } = 177.067 – 167.555 = 9.513 \)
\( \phi_{R} = 1 \)
\( \phi_{T} = n – 1 = 19 – 1 =18 \)
\( \phi_{E} = \phi_{T} – \phi_{R} = 18 – 1 = 17 \)
\( V_{R} = \displaystyle\frac{S_{R}}{\phi_{R}} = \displaystyle\frac{ 167.555 }{ 1 } = 167.555 \)
\( V_{E} = \displaystyle\frac{S_{E}}{\phi_{E}} = \displaystyle\frac{ 9.513 }{ 17 } = 0.560 \)
\( 分散比 = \displaystyle\frac{V_{R}}{V_{E}} = \displaystyle\frac{ 167.555 }{ 0.560 } = 299.433 \)
\( F(1,17:0.05) = 4.45 \)
\( 分散比 = \displaystyle\frac{V_{R}}{V_{E}} = 299.433 > F(\phi_{R}, \phi_{E}:0.05) = F(1,17:0.05) = 4.45 \)
\( \mu = 0 \) \( \sigma^2 = 1 \) \( N( 0 , 1^2 ) \)
\( Pr( X = x ) \) \( Pr( X = x ) = {}_n \mathrm{ C }_x P^{ x } ( 1 – P)^{ n – x } = \displaystyle\frac{ n! }{ x! ( n – x )! } P^{ x } ( 1 – P )^{ n – x } \)
\( {}_n \mathrm{ C }_x \) \( {}_n \mathrm{ C }_x = \displaystyle\frac{ n! }{ x! ( n – x )! } \)
\( E( X ) = nP \) \( D( X ) = \sqrt{ n P ( 1 – P ) } \)
\( p = X/n \) \( E( p ) = P \) \( D( p ) = \sqrt{ \displaystyle\frac{ P ( 1 -P ) }{ n } } \)
\( nP \geq 5 , n ( 1-P ) \geq 5 \)
\( np = \lambda \)
\( Pr( X = k ) = \displaystyle\frac{ e^{ – \lambda } \lambda ^k }{ k! } ( k = 0, 1, 2, \cdots ) \)
\( E( k ) = \lambda \) \( D( k ) = \sqrt{ \lambda } \)
\( Pr( u \geq K_{ p } ) = P \)
\( U = \displaystyle\frac{ X – \mu }{ \sigma } \)
\( N( 20, 4^2 ) \)
\( U = \displaystyle\frac{ 24 – 20 }{ 4 } = 1 \)
\( N( 100, 30^2 ) \)
\( U = \displaystyle\frac{ 53.5 – 100 }{ 30 } = 1.55 \)
\( S(xy) = \displaystyle \sum_{ i=1 }^n ( x_{ i } – \bar{ x })( y_{ i } – \bar{ y }) \)
\( S(xy) = \displaystyle \sum_{ i=1 }^n x_{ i }y_{ i } – n \sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } \)
\( S(xy) = \displaystyle \sum_{ i=1 }^n x_{ i }y_{ i } – \displaystyle\frac{\sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } }{ n } \)
\( S(xy) = – \displaystyle\frac{\sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } }{ n } \)
\( F( \phi_{ R }, \phi_{ E } : 0.05 ) \)
\( F(\phi_{R}, \phi_{E}:0.05) \)
\( CL = \bar{ X } \) \( UCL = \bar{ X } + 2.660 \bar{ Rs } \) \( LCL = \bar{X} – 2.660 \bar{ Rs } \)
\( CL = \bar{ Rs } \) \( UCL = 3.267 \bar{ Rs } \)
\( CL = \bar{ s } \) \( UCL = B_{ 4 } \bar{ s } \) \( LCL = B_{ 3 } \bar{ s } \)
\( CL = \bar{ c } \) \( UCL = \bar{ c } + 3 \sqrt{ \bar{ c }} \) \( LCL = \bar{ c } – 3 \sqrt{ \bar{ c }} \)
\( CL = \bar{ u } \) \( UCL = \bar{ u } + 3 \sqrt{ \displaystyle\frac{ \bar{ u }}{ n} } \) \( LCL = \bar{ u } – 3 \sqrt{ \displaystyle\frac{ \bar{ u }}{ n} } \)