Question

Boyle's law If you have taken a chemistry or physics class, then you are probably familiar with Boyle's law: for gas in a confined space kept at a constant temperature, pressure times volume is a constant (in symbols, $\mathrm{PV}=\mathrm{k}PV=k$). Students in a chemistry class collected data on pressure and volume using a syringe and a pressure probe. If the true relationship between the pressure and volume of the gas is $\mathrm{PV}=\mathrm{k},PV=k$, then

$\mathrm{P}=\mathrm{k}1\mathrm{V}P=k\frac{1}{V}$

Here is a graph of pressure versus a volume, $\frac{1}{\text{volume}}$, along with output from a linear regression analysis using these variables:

a. Give the equation of the least-squares regression line. Define any variables you use.

b. Use the model from part (a) to predict the pressure in the syringe when the volume is $17$cubic centimeters.

Short answer

a). The equation of the least-squares regression line is $\hat{y}=0.36774+15.8994\times \frac{1}{x}$.

b). The expected pressure is $1.3030\mathrm{atm}$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

Least square regression line's general equation

$\hat{y}=b_0+b_{1}x$

In the row "constant" and the column "Coef" of the computer output, the calculated constant $b_0$is indicated.

$b_0=0.36774$

In the row $"1/V"$and the column "Coef" of the computer output, the slope $b_1$is calculated.

$b_1=15.8994$

Part (a) Step 3: Explanation

Substituting the value of $b_0$and $b_1$

$\hat{y}=b_0+b_{1}x$

$\hat{y}=0.36774+15.8994x$

The reciprocal of the volume is $x$, and the pressure is $y$.

role="math" localid="1654252842611" $\hat{y}=0.36774+15.8994\times \frac{1}{x}$

Where $y$ denotes pressure and $x$ denotes volume.

Part (b) Step 1: Given Information

Given data:

Part (b) Step 2: Explanation

The least-squares regression line's equation is:

$\hat{y}=0.36774+15.8994\times \frac{1}{x}$

$x$denotes volume, while $y$denotes pressure.

Substituting the value of $x$:

$\hat{y}=0.36774+15.8994\times \frac{1}{17}$

$\hat{y}=1.3030$