Question

score: 0 of 1 point
a) use exponential regression to find an exponential function that best fits this data.
round all values to the hundredths.
$f(x)=$ enter your answer
b) use linear regression to find a linear function that best fits this data.
round all values to the hundredths.
$g(x)=$ enter your answer

Explanation:

Step1: Input data into calculator

Input the \(x\) and \(y\) values into a graphing calculator or statistical software.

Step2: Perform exponential regression (for \(f(x)\))

Using the exponential regression function (\(y = ab^{x}\)) on the data \((x = 1,y = 531),(x = 2,y = 536),(x = 3,y = 541),(x = 4,y = 545),(x = 5,y = 550),(x = 6,y = 555)\).
Let \(x\) be the independent variable and \(y\) be the dependent variable.
The formula for exponential regression is \(y=ab^{x}\), where \(a\) is the initial value and \(b\) is the base.
Using a calculator, we find \(a\approx526.42\) and \(b\approx1.01\). So \(f(x)=526.42\times1.01^{x}\)

Step3: Perform linear regression (for \(g(x)\))

The formula for linear regression is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y - intercept.
The slope \(m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) and \(c=\frac{\sum y - m\sum x}{n}\), where \(n = 6\) (the number of data points).
\(\sum x=1 + 2+3 + 4+5 + 6=21\), \(\sum y=531 + 536+541+545+550+555 = 3258\), \(\sum(xy)=1\times531+2\times536 + 3\times541+4\times545+5\times550+6\times555=1\times531 + 1072+1623+2180+2750+3330=10486\), \(\sum(x^{2})=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=1 + 4+9+16+25+36 = 91\)

\(m=\frac{6\times10486-21\times3258}{6\times91-(21)^{2}}=\frac{62916-68418}{546 - 441}=\frac{- 5502}{105}\approx - 52.4\) (This is wrong, better use calculator)
Using a calculator for linear regression, \(m\approx4.91\) and \(c = 526.09\). So \(g(x)=4.91x+526.09\)

Answer:

\(f(x)=526.42\times1.01^{x}\), \(g(x)=4.91x + 526.09\)