Chapter 12.1, Problem 20E

To determine

To find: that is there convincing evidence of a linear relation between number of pencils for students at this high school and GPA.

Answer to Problem 20E

No

Explanation of Solution

Given:

  

  $\begin{array}{l}n=\text{unknown}\\ \alpha \text{=Significance level =0}\text{.05}\end{array}$

Claim: Slope is negative (hurts performance)

Formula used:

The test statistics

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}$

Calculation:

The calculation of the slope $b_1$ is mention in the row “Pencils” and in the column “Coef” of the mention in the computer output:

  $b_1=-0.0423$

The calculated standard deviation of the slope $S{E}_{b_1}$ is mention in the row “Pencils” and in the column “SE Coef” of the mention in the computer output:

  $S{E}_{b_1}=0.0631$

The null hypothesis or the alternative hypothesis statement is the mention in the claim. The null hypothesis statement is that the slope is zero. If the mention claim is the null hypothesis, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Test statistic:

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}=\frac{-0.0423-0}{0.0631}=-0.6704$

The P-value is mention in the row “Pencils” and in the column “P” of the computer output= $P=0.5062$

If the P-value is equal or less than to the significance level, then the null hypothesis is rejected:

  $P>0.05\Rightarrow \text{Fail to reject }{H}_0$

There is enough sufficient proof of a linear relationship between number of pencils for students at this high school and GPA.