91影视

The 95% confidence intervals for the regression coefficients \({\beta _1}\) and \({\beta _0}\)need to be constructed. The sample size (n) is equal to 153. The number of predictors (k) is equal to 2. The degrees of freedom are computed using the formula n-(k+1). The critical value of t is given to be equal to 1976.

Margin of error corresponding to\({\beta _1}\):

Let\({b_1}\)be the estimated value of\({\beta _1}\).

Using the output from exercise 17, the value of standard error of\({b_1}\)is equal to :\({s_{{b_1}}} = 0.0711\). The value of\({b_1}\)is equal to 0.7693.

It is given that\({t_{\frac{\alpha }{2}}} = 1.976\).

Substituting the required values in the formula of E, the following of E is obtained:

\(\begin{array}{c}{E_1} = {t_{\frac{\alpha }{2}}}{s_{{b_1}}}\\ = 1.976 \times 0.0711\\ = 0.1405\end{array}\)

Therefore, the margin of error is 0.1405.

Margin of error corresponding to\({\beta _1}\):

Let\({b_2}\)be the estimated value of\({\beta _2}\).

Using the output from exercise 17, the value of standard error of\({b_2}\)is equal to\({s_{{b_2}}} = 0.0338\). The value of\({b_2}\)is equal to 1.0095.

It is given that\({t_{\frac{\alpha }{2}}} = 1.976\).

Substituting the required values in the formula of E, the following of E is obtained:

\(\begin{array}{c}{E_2} = {t_{\frac{\alpha }{2}}}{s_{{b_1}}}\\ = 1.976 \times 0.0338\\ = 0.0668\end{array}\)

Therefore, the margin of error is 0.0668.

Confidence interval for\({\beta _1}\):

The 95% confidence interval for\({\beta _1}\)is computed as follows:

\(\begin{array}{c}{b_1} - {E_1} < {\beta _1} < {b_1} + {E_1}\\0.7693 - 0.1405 < {\beta _1} < 0.7693 + 0.1405\\0.6288 < {\beta _1} < 0.9098\end{array}\)

Therefore, the 95% confidence interval of\({\beta _1}\)is (0.6288,0.9098).

Confidence interval for\({\beta _2}\):

The 95% confidence interval for\({\beta _2}\)is computed as follows:

\(\begin{array}{c}{b_2} - {E_2} < {\beta _2} < {b_2} + {E_2}\\1.0095 - 0.0668 < {\beta _2} < 1.0095 + 0.0668\\0.9427 < {\beta _2} < 1.0763\end{array}\)

Therefore, the confidence interval of \({\beta _2}\) is equal to (0.9427,1.0763).

Let\({\beta _1}\)be the regression coefficient of the predictor 鈥淗eight鈥�.

The confidence interval of\({\beta _1}\)is equal to (0.6288, 0.9098).

Since the confidence interval does not contain the value 0, the variable 鈥淗eight鈥� is significant in predicting the variable 鈥淲eight鈥�.

Let\({\beta _2}\)be the regression coefficient of the predictor 鈥淲aist鈥�.

The confidence interval of\({\beta _1}\)is equal to (0.9427, 1.0763).

Since the confidence interval does not contain the value 0, the variable 鈥淲aist鈥� is significant in predicting the variable 鈥淲eight鈥�.