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• We need to use MINITAB method of technology to findif it makes sense to apply the regression t-test .

• An industrial pollutant, polychlorinated biphenyls (PCBs), are carcinogenic and are known to be a major threat to natural ecosystems.
• One study is entitled "Effects of Environmental Pollution Substances on Non-Human Animals". Sixtyannouncer pelican eggs were collected and the shell thickness (millimeters (mm)) and PCB concentration (parts per million (ppm)) were measured.
• The data can be found on the Weiss Stats website.

The following are the procedures of MINITAB method to construct the residual plot.

• Step 1: Choose Stat > Regression > Regression.
• Step 2: In Response, enter the column THICKNESS.
• Step 3: In Predictors, enter the columns PCB.
• Step 4: In Graphs, enter the columns PCB under Residuals versus the variables.
• Step 5: Click OK.

The following are the output will come when we are using MINITAB method.

The following are the procedures of MINITAB method to construct the normal probability plot of residuals.

• Step 1: Choose Stat > Regression > Regression.
• Step 2: In Response, enter the column THICKNESS
• Step 3: In Predictors, enter the columns PCB.
• Step 4: In Graphs, select Normal probability plot of residuals.
• Step 5: Click OK.

The following are the Output of Construct the normal probability plot of residuals by using MINITAB method

The following are the the assumption for regression inferences

• Population regression line:

The conditional mean of the response variable $\left(Y\right)$is ${\mathrm{β}}_0+{\mathrm{β}}_{1}X$, for each value $X$of predicator variable.

• Equal standard deviation:

The standard deviation for the response variable $\left(Y\right)$is same for the standard deviation for the explanatory variable $\left(X\right)$. The standard deviation is denoted as $\mathrm{σ}$.

• Normal populations:

The distribution of responsevariables usually follows.

• Independent observations:

Theobserved values of the response are independent of each other.

• From the residual plot, it is clear that the residuals are in the horizontal band.
• The normal probability plot of the residuals shows that the residuals are displayed in a linear pattern.

This does not violate assumption 1-3of the variable thickness and the PCB regression conclusion.

• We are using MINITAB method of technology for finding specific data that provide sufficient evidence to conclude that predictors are useful in predicting response variables.

• An industrial pollutant, polychlorinated biphenyls (PCBs), are carcinogenic and are known to be a major threat to natural ecosystems.
• One study is entitled "Effects of Environmental Pollution Substances on Non-Human Animals". Sixty announcer pelican eggs were collected and the shell thickness (millimeters (mm)) and PCB concentration (parts per million (ppm)) were measured.
• The data can be found on the Weiss Stats website.

The following are the appropriate hypothesis to find the given data provides enough evidence to conclude that PCB is useful for predicting THICKNESS.

• Null hypothesis:

$H_0:{\mathrm{β}}_1=0$

That is, the data do not provide sufficient evidence to conclude that PCB are useful in predicting THICKNESS.

• Alternative hypothesis:

$H_a:{\mathrm{β}}_1≠0$

That is, the data provide sufficient evidence to conclude that PCB are useful in predicting THICKNESS.

• Rejection Rule:

If $p$-value $≤\mathrm{Î\pm }(=0.05)$, then reject the null hypothesis $\left(H_0\right)$.

The following are the MINITAB procedure to Find the test statistic and $p$-value:

• Step 1: Choose Stat > Regression > Regression.
• Step 2: In Response, enter the column THICKNESS.
• Step 3: In Predictors, enter the columns PCB.
• Step 4: Click OK.

Regression Analysis: THICKNESS versus PCB

The regression equation is

THICKNESS$=0.364-0.000201\mathrm{PCB}$

$\begin{array}{lrrrr}\text{Predictor}& \text{Coef}& \text{SE Coef}& T& \text{P}\\ \text{Constant}& 0.36392& 0.03042& 11.96& 0.000\\ \text{PCB}& -0.0002014& 0.0001404& -1.43& 0.157\end{array}$

$S=0.0716205R-Sq=3.4\%R-Sq\left(adj\right)=1.8\%$

From the MINITAB output, the value of test statistic is $-1.43$and the $p$-value is $0.157$.

Conclusion:

• Use the significance level, $\mathrm{Î\pm }=0.05$.
• Here, $p$-value is greater than the level of significance.
• That is, $p$-value $(=0.157)>\mathrm{Î\pm }(=0.05)$.
• Therefore, by the rejection rule, it can be concluded that there is no evidence to reject the null hypothesis $\left(H_0\right)$at $\mathrm{Î\pm }=0.05$.

That is, the data does not provide sufficient evidence to conclude that PCBare useful in predicting THICKNESS.