Chapter 12, Problem 48CE

a.

To determine

Check whether the 95% confidence interval for the slope include zero. If so explain it and if not explain it.

Answer to Problem 48CE

The 95% confidence intervaldoes not contain zero.

Explanation of Solution

Answer will vary.

Here the data set B is taken, in which the midterm exam and final exam score is given.

Regression:

Suppose $x_1\mathrm{...}{x}_n$ be n sample values of independent variable and the corresponding dependent variable values are $y_1\mathrm{...}{y}_n$. The slope and the intercept of ordinary least square can be defined as $b_0=\overline{y}-b_1\overline{x}$ and $b_1=\frac{S{S}_{xy}}{S{S}_{xx}}$.

Where, $S{S}_{xx},S{S}_{yy},S_{xy}$ are the sum of squares due to x, y andxy respectively. $\overline{x}\text{and}\overline{y}$ are the sample mean of the independent and dependent variable respectively.

The total sum of squares is denoted as, $SST={\displaystyle \sum _{i=1}^n\left(y_i-\overline{y}\right)}$.

The regression sum of squares is denoted as, $SSR={\displaystyle \sum _{i=1}^n\left({\widehat{y}}_i-\overline{y}\right)}$.

The error sum of squares is denoted as, $SSE={\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$.

From the regression the fitted line is denoted as, $\widehat{y}=b_0+b_{1}x$.

The 95% confidence interval for the slope, ${\beta }_1$ is defined as,

$b_1-t_{\frac{\alpha }{2}}{s}_{b_1}\le {\beta }_1\le b_1+t_{\frac{\alpha }{2}}{s}_{b_1}$

Where, $b_1$ is the slope, $t_{\frac{\alpha }{2}}$ be the table value of t-distribution for two-tailed with $\alpha$ level of significance and $s_{b_1}$ is the standard error of the slope denoted as, $s_{b_1}=\frac{s_e}{\sqrt{{\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)}}},s_e\text{ is the standard error}\text{.}$$s_e=\sqrt{\frac{SSE}{n-2}}$ with sample size n and the sum of squares due to error.

Procedure for 95% confidence interval using EXCEL:

Software Procedure:

Step-by-step software procedure to find 95% confidence interval using EXCEL software is as follows:

• • Open an EXCEL file.
• • In column A and B, the Midterm Exam Score and Final Exam Score data were entered.
• • Click on data > click on Data analysis.
• • Choose Regression > click OK.
• • Select Input Y range asthe column ofFinal Exam Score.
• • Select Input X range asthe column ofMidterm Exam Score.
• • Select the output range.
• • Click OK.
• Regression using EXCEL software is given below:

Thus, the 95% confidence interval of the slope is (0.7012, 1.3263).

Interpretation:

From the confidence interval it can be concluded that there is 95% confident that the slope will lie between 0.7012 and 1.3263.

The interval does not contain zero.

Theslope is not zero as the confidence interval does not contain zero.

b.

To determine

Perform a two-tailed t test for zero slope at $\alpha =0.05$.

State the hypotheses, degrees of freedom and critical value for the test.

Answer to Problem 48CE

There is nosufficient evidence to support that the slope is zero.

The hypotheses are:

Null hypothesis:

$H_0:{\beta }_1=0$

That is, the slope is zero.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$

That is, the slope not equal to zero.

The degrees of freedom is56.

The critical value is $\pm 2.00324$ .

Explanation of Solution

Calculation:

Hypotheses:

Null hypothesis:

$H_0:{\beta }_1=0$

That is, the slope is zero.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$

That is, the slope is not equal to zero.

From part(a), the test statistics is 6.498.

Critical value:

Here, the sample size, $n=58$.

The degrees of freedom is,

$\begin{array}{c}df=n-2\\ =58-2\\ =56\end{array}$

Thus, the degrees of freedom is56.

For two tailed test, the critical value for t-test will be, $t_{\frac{\alpha }{2},\left(n-2\right)}$.

It is assumed that the level of significance, $\alpha =0.05$.

Procedure for critical-value using EXCEL:

Software Procedure:

Step-by-step software procedure to obtain critical-value using EXCEL software is as follows:

• • Open an EXCEL file.
• • In cell A1, enter the formula “=T.INV(0.025,56)”
• Output using EXCEL software is given below:

Here, the test is a two tailed test. Hence, the critical value will be $\pm 2.00324$ .

The level of significance is 0.05.

Decision rule:

If $t_{cal}>t_{\alpha ,\left(n-2\right)}$ , reject the null hypothesis.

If $t_{cal}\le t_{\alpha ,\left(n-2\right)}$ , fail to reject the null hypothesis.

Conclusion:

Here the test statistic value is greater than the critical value.

That is, $t_{cal}\left(=6.498\right)>t_{0.05,56}\left(=2.003\right)$ .

Hence, by the decision rule, reject the null hypothesis.

That is, the slope differs from zero.

Therefore, it can be concluded that there is nosufficient evidence to support that the slope is zero.

c.

To determine

Interpret the p-value for the slope.

Explanation of Solution

From the output in part (a) it is observed that, the p-value for the slope is 0.000. That is, if midterm and final scores are not correlated then there is very little chance of observing this sample.

d.

To determine

Check whether the sample support the hypothesis about the sign of the slope.

Explanation of Solution

From part (b), there is no sufficient evidence to support that the slope is zero. Thus, the slope is either positive or negative. Hence, the sample supports the hypothesis about the sign of the slope.