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The value of the measures of pain intensity before Duragesic treatment and after Duragesic treatment are considered.

Let x represent thepain intensity before Duragesic treatment

Let y represent the painintensity afterDuragesic treatment

Here, y is the response variable, and x is the predictor variable

The regression line of y on x has the following expression:

\(\hat y = {b_0} + {b_1}x\)where

\({b_0}\) is the intercept term

\({b_1}\) is the slope coefficient

The following table shows the required computations to compute the regression line:

The slopeof the regression lineis computed as follows:

\(\begin{aligned} {b_1} &= \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ &= \frac{{31\left( {533.78} \right) - \left( {144.9} \right)\left( {102.2} \right)}}{{31\left( {866.35} \right) - {{\left( {144.9} \right)}^2}}}\\ &= 0.2966\end{aligned}\)

Therefore, the value of the slope coefficient is 0.2966.

The interceptis computed as follows:

\(\begin{aligned} {b_0} &= \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ &= \frac{{\left( {102.2} \right)\left( {866.35} \right) - \left( {144.9} \right)\left( {533.78} \right)}}{{31\left( {866.35} \right) - {{\left( {144.9} \right)}^2}}}\\ &= 1.9103\end{aligned}\)

Therefore, the value of the intercept is 1.9103.

Theregression equationis given as follows:

\(\begin{aligned} \hat y &= {b_0} + {b_1}x\\ &= 1.9103 + 0.2966x\end{aligned}\)

Thus, the regression equation is \(\hat y = 1.9103 + 0.2966x\).

If Duragesic treatment has no effect, then the values of the pain intensities before and after the treatment will be equal.

In the given data, the following two pairs of data have the same pain intensities before and after the treatment:

Now, the modified slope coefficient is equal to:

\(\begin{aligned}{c}{b_1} &= \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ &= \frac{{2\left( {27.60} \right) - \left( {13.8} \right)\left( 4 \right)}}{{2\left( {95.22} \right) - {{\left( {13.8} \right)}^2}}}\\ &= 0\end{aligned}\)

The modified intercept term is equal to:

\(\begin{aligned} {b_0} &= \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ &= \frac{{\left( 4 \right)\left( {95.22} \right) - \left( {13.8} \right)\left( {27.60} \right)}}{{4\left( {95.22} \right) - {{\left( {13.8} \right)}^2}}}\\ &= 2.00\end{aligned}\)

Thus, the regression equation when the treatment has no effect is as follows:

\(\hat y = 2.00 + 0x\)