Question

an unknown metal has been found and the following experimental results have been tabulated in the table below. the table contains the grams of the unknown metal and the volume in milliliters of water displacement. find a linear model that expresses the volume as a function of grams. round your answers to 3 decimal places

grams	volume in ml
23.5	169.3
26	181.8
28.5	199.3
31	219
33.5	236.6
36	254.3

predicted volume (y)= \\(\square x + \square\\) where x is the grams of the unknown metal

Explanation:

Step1: Calculate the mean of \(x\) (grams) and \(y\) (volume in ml)

Let \(x=\{21,23.5,26,28.5,31,33.5,36\}\) and \(y =\{152.8,169.3,181.8,199.3,219,236.6,254.3\}\)
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{21 + 23.5+26+28.5+31+33.5+36}{7}=\frac{199.5}{7}=28.5\)
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{152.8+169.3+181.8+199.3+219+236.6+254.3}{7}=\frac{1413.1}{7}=201.8714\)

Step2: Calculate the slope \(b\)

\(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_{1}-\bar{x})(y_{1}-\bar{y})=(21 - 28.5)(152.8-201.8714)=(- 7.5)(-49.0714) = 368.0355\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(23.5 - 28.5)(169.3 - 201.8714)=(-5)(-32.5714)=162.857\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(26 - 28.5)(181.8 - 201.8714)=(-2.5)(-20.0714) = 50.1785\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(28.5 - 28.5)(199.3 - 201.8714)=0\times(-2.5714)=0\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(31 - 28.5)(219 - 201.8714)=2.5\times17.1286 = 42.8215\)
\((x_{6}-\bar{x})(y_{6}-\bar{y})=(33.5 - 28.5)(236.6 - 201.8714)=5\times34.7286=173.643\)
\((x_{7}-\bar{x})(y_{7}-\bar{y})=(36 - 28.5)(254.3 - 201.8714)=7.5\times52.4286 = 393.2145\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=368.0355+162.857+50.1785 + 0+42.8215+173.643+393.2145=1190.75\)
\((x_{1}-\bar{x})^{2}=(21 - 28.5)^{2}=56.25\)
\((x_{2}-\bar{x})^{2}=(23.5 - 28.5)^{2}=25\)
\((x_{3}-\bar{x})^{2}=(26 - 28.5)^{2}=6.25\)
\((x_{4}-\bar{x})^{2}=(28.5 - 28.5)^{2}=0\)
\((x_{5}-\bar{x})^{2}=(31 - 28.5)^{2}=6.25\)
\((x_{6}-\bar{x})^{2}=(33.5 - 28.5)^{2}=25\)
\((x_{7}-\bar{x})^{2}=(36 - 28.5)^{2}=56.25\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=56.25+25+6.25+0+6.25+25+56.25 = 175\)
\(b=\frac{1190.75}{175}=6.804\)

Step3: Calculate the intercept \(a\)

\(a=\bar{y}-b\bar{x}\)
\(a = 201.8714-6.804\times28.5\)
\(a=201.8714 - 193.914=7.957\)

Answer:

Predicted Volume \((\hat{y})=6.804x + 7.957\)