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The pain intensities of a group of subjects are recorded before using the drug Duragesic and after using the drug Duragesic.

Follow the given steps to construct a scatterplot between the pain intensities before and after the treatment:

• Mark the values 0, 2, 4, 鈥︹��., 12 on the horizontal scale and label the axis as 鈥淧ain Intensity Before Treatment鈥�.
• Mark the values 0, 1, 2, 鈥︹��., 10 on the vertical scale and label the axis as 鈥淧ain Intensity After Treatment鈥�.
• Plot points on the graph for each value of the pain intensity after treatment for the corresponding value on the pain intensity before treatment.

The following scatterplot is constructed:

The points on the plot appear to be scattered randomly, indicating that there is not a significant linear correlation between the pain intensities before and after the treatment.

Let x denote the values of the pain intensity before treatment.

Let y denote the values of the pain intensity after treatment.

The formula for computing the correlation coefficient (r) between the values of x and y is as follows:

\(r = \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\)

The following calculations are done to compute the value of r:

Substituting the above values, the following value of r is obtained:

\(\begin{aligned} r &= \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\\ &= \frac{{31\left( {533.78} \right) - \left( {144.9} \right)\left( {102.2} \right)}}{{\sqrt {31\left( {866.35} \right) - {{\left( {144.9} \right)}^2}} \sqrt {31\left( {516.4} \right) - {{\left( {102.2} \right)}^2}} }}\\ &= 0.304\end{aligned}\)

Therefore, the value of r is equal to 0.304.

Here, n=31.

If the value of the correlation coefficient lies between the critical values, then the correlation between the two variables is considered insignificant else, it is considered significant.

The critical values of r for n=31 and\(\alpha = 0.05\)are -0.355 and 0.355.

The corresponding p-value of r is equal to 0.0964.

Since the computed value of r equal to 0.304 lies between the values -0.355 and 0.355, and the p-value is greater than 0.05, it can be said that the correlation between the two variables is not significant.

Therefore, the linear correlation between the pain intensities before and after the treatment is not significant.

Even if there is a significant linear correlation between the pain intensities before and after the treatment, it could not be concluded that the drug is effective in reducing pain. The linear correlation only tells us about the association between the two variables and not about the difference in the values of the two variables and whether the overall value of one variable is lower than the other variable.

A separate kind of hypothesis test needs to be conducted to conclude the effectiveness of the drug.