Question

11:50
94
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 mass as a function of
the volume.
grams volume in ml
13 148.9
15.5 174
18 200
20.5 234.8
23 266
25.5 292.1
28 311.2
a. write the linear regression equation for the data in
the chart where x is the grams of the unknown
metal. first the slope, then the y - intercept. round
each answers to 3 decimal places:
volume = x+

b. if the mass of an unknown metal is 11, using your
un - rounded regression equation find its predicted
volume (in). round your answer to 1 decimal
place. ml
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Explanation:

Step1: Calculate the slope ($m$)

Let $x$ be grams and $y$ be volume.
Using the formula for slope \(m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\)
where \(n = 7\) (number of data points)
\(\sum x=13 + 15.5+18+20.5+23+25.5+28=143.5\)
\(\sum y=148.9+174+200+234.8+266+292.1+311.2 = 1627\)
\(\sum(xy)=13\times148.9+15.5\times174+18\times200+20.5\times234.8+23\times266+25.5\times292.1+28\times311.2\)
\(=1935.7+2707+3600+4813.4+6118+7448.55+8713.6=35336.25\)
\(\sum(x^{2})=13^{2}+15.5^{2}+18^{2}+20.5^{2}+23^{2}+25.5^{2}+28^{2}\)
\(=169+240.25+324+420.25+529+650.25+784 = 3116.75\)
\(m=\frac{7\times35336.25-143.5\times1627}{7\times3116.75-(143.5)^{2}}\)
\(=\frac{247353.75 - 233474.5}{21817.25 - 20592.25}=\frac{13879.25}{1225}\approx11.329\)

Step2: Calculate the y - intercept ($b$)

Using the formula \(b=\bar{y}-m\bar{x}\)
\(\bar{x}=\frac{\sum x}{n}=\frac{143.5}{7}=20.5\)
\(\bar{y}=\frac{\sum y}{n}=\frac{1627}{7}\approx232.429\)
\(b = 232.429-11.329\times20.5\)
\(b=232.429 - 232.2445=0.1845\approx0.185\)

Step3: For part B

The regression equation is \(y = 11.329x+0.185\) (un - rounded equation: \(y = 11.329459x+0.1845\))
When \(x = 11\)
\(y=11.329459\times11+0.1845\)
\(y=124.624049+0.1845=124.808549\approx124.8\)

Answer:

A. Volume \(=11.329x + 0.185\)
B. \(124.8\)