91影视

Values are given on two variables namely, actress age and actor age.

Let x represent the age of Best Actress.

Let y represent the age of Best Actor.

Themean value of xis given as,

\(\begin{array}{c}\bar x = \frac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}\\ = \frac{{28 + 30 + .... + 54}}{{12}}\\ = 39.083\end{array}\)

Therefore, the mean value of x is 39 years.

Themean value of yis given as,

\(\begin{array}{c}\bar y = \frac{{\sum\limits_{i = 1}^n {{y_i}} }}{n}\\ = \frac{{43 + 37 + .... + 33}}{{12}}\\ = 45.167\end{array}\)

Therefore, the mean value of y is 45 years.

The standard deviation of x is given as,

\(\begin{array}{c}{s_x} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( {28 - 39.083} \right)}^2} + {{\left( {30 - 39.083} \right)}^2} + ..... + {{\left( {54 - 39.083} \right)}^2}}}{{12 - 1}}} \\ = 13.734\end{array}\)

Therefore, the standard deviation of x is 13.734.

The standard deviation of yis given as,

\(\begin{array}{c}{s_y} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({y_i} - \bar y)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( {43 - 45.167} \right)}^2} + {{\left( {37 - 45.167} \right)}^2} + ..... + {{\left( {33 - 45.167} \right)}^2}}}{{12 - 1}}} \\ = 7.872\end{array}\)

Therefore, the standard deviation of y is 7.872.

Thecorrelation coefficientis given as,

\(r = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {\left( {\left( {n\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \right)\left( {\left( {n\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} \right)} }}\)

The calculations required to compute the correlation coefficient are as follows:

The correlation coefficient is given as,

\(\begin{array}{l}r = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {\left( {\left( {n\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \right)\left( {\left( {n\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} \right)} }}\\ = \frac{{12\left( {20841} \right) - \left( {469} \right)\left( {542} \right)}}{{\sqrt {\left( {\left( {12 \times 20405} \right) - {{\left( {469} \right)}^2}} \right)\left( {\left( {12 \times 25162} \right) - {{\left( {542} \right)}^2}} \right)} }}\\ = - 0.288\end{array}\)

Therefore, the correlation coefficient is -0.288.

The slopeof the regression line is given as,

\(\begin{array}{c}{b_1} = r\frac{{{s_Y}}}{{{s_X}}}\\ = - 0.288 \times \frac{{7.872}}{{13.734}}\\ = - 0.165\end{array}\)

Therefore, the value of slope is - 0.165.

The interceptis computed as,

\(\begin{array}{c}{b_0} = \bar y - {b_1}\bar x\\ = 45.167 - \left( { - 0.165 \times 39.083} \right)\\ = 51.6\end{array}\)

Therefore, the value of intercept is 51.6.

Theregression equationis given as,

\(\begin{array}{c}\hat y = {b_0} + {b_1}x\\ = 51.6 - 0.165x\end{array}\)

Thus, the regression equation is \(\hat y = 51.6 - 0.165x\).

Referring to exercise 21 of section 10-1 for the following result

1)The scatter plot shows approximately a linear relationship between the variables.

2)The P-value is 0.364.

As the P-value is greater than the level of significance (0.05), this implies the null hypothesis fails to reject.

Therefore, the correlation is not significant.

Referring to Figure 10-5, thecriteria for a good regression model are not satisfied.

Therefore, the regression equation cannot be used to predict the value of y.