Question

Question:For the conditions of Exercise \({\bf{7}}\), and the data in Table \({\bf{11}}{\bf{.1}}\), carry out a test of the following hypotheses.

\(\begin{array}{*{20}{c}}{{{\bf{H}}_{\bf{0}}}{\bf{:}}{{\bf{\beta }}_{\bf{2}}}{\bf{ = 0}}}\\{{{\bf{H}}_{\bf{1}}}{\bf{:}}{{\bf{\beta }}_{\bf{2}}} \ne {\bf{0}}}\end{array}\)

Short answer

The value of the test statistic is \({U_2} = 0.095\)with a value of \(0.927\). So we cannot reject the null hypothesis.

Step-by-step solution

Define linear statistical models

A linear model describes the correlation between the dependent and independent variables as a straight line.

\(y = {a_0} + {a_1}{x_1} + {a_2}{x_2} + \tilde A,\hat A1/4 + {a_n}{x_n}\)

Models using only one predictor are simple linear regression models. Multiple predictors are used in multiple linear regression models. For many response variables, multiple regression analysis models are used.

Find the value of test statistic

For the given data we’ve fitted the regression function,

\(y = - 0.745 + 0.619x + 0.013{x^2}\)

Now the test statistic used for the given hypothesis is

\({U_2} = \frac{{{{\hat \beta }_2} - \beta _2^ * }}{{\sigma ' \cdot \sqrt {{\xi _{22}}} }}\mathop \sim \limits^{} {t_{n - p}}\)

Calculate the estimated standard variance.

This follows a Student’s \(t\)-distribution with degrees of freedom\(n - p = 10 - 3 = 7\).

Therefore the test statistic is

\(\begin{array}{c}{U_2} = \frac{{0.013 - 0}}{{1.157 \cdot \sqrt {0.014} }}\\ = 0.095\end{array}\).

Hence the test statistic is \(0.095\).

Testing the hypothesis

Since the test is two-sided, then the \(p - \)value of this test is the doubled area under the density curve of the student’s \(t\)-distribution.

To find the approximate \(p - \)value use the Table.

From the table and the test statistic we got\(0.095\), which is greater than \(0.005\).

We can conclude that,

\(p{\rm{ - value\;}} > 0.90\)

From statistical software we got,

\(p{\rm{ - value\;}} = 0.927\)

Since the \(p - \)value is greater than \(0.005\)we cannot reject the null hypothesis.