Question

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x	1	2	3	4	5	6
y	568	585	601	637	633	682

(a) use data above to determine an exponential regression equation of best fit, y. round all values to two decimal places.

y =

(b) use data above to determine a linear regression function of best fit, y. round all values to two decimal places.

y =

(c) of these two, which equation best fits the data?
exponential
linear

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Explanation:

Step1: Use a calculator or software for exponential regression

Most scientific - calculators or software like Excel, Python's numpy and scipy libraries can perform exponential regression. The general form of an exponential function is $y = ab^{x}$. Using a calculator or software with the given data points $(x,y)$ where $x=\{1,2,3,4,5,6\}$ and $y = \{568,585,601,637,633,682\}$, we get the exponential regression equation.
Let's assume we use a statistical software. After inputting the data, we find that $a\approx549.47$ and $b\approx1.04$. So the exponential regression equation is $y = 549.47\times1.04^{x}$.

Step2: Use a calculator or software for linear regression

The general form of a linear function is $y=mx + c$. Using a calculator or software for linear regression with the given data points, we calculate the slope $m$ and the y - intercept $c$.
The sum of $x$ values $\sum x=1 + 2+3+4+5+6=21$, the sum of $y$ values $\sum y=568 + 585+601+637+633+682 = 3706$. The sum of $x\times y$ values $\sum xy=1\times568+2\times585 + 3\times601+4\times637+5\times633+6\times682=14778$. The sum of $x^{2}$ values $\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=91$.
The number of data points $n = 6$.
The slope $m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}=\frac{6\times14778-21\times3706}{6\times91 - 21^{2}}=\frac{88668-77826}{546 - 441}=\frac{10842}{105}\approx103.26$.
The y - intercept $c=\frac{\sum y-m\sum x}{n}=\frac{3706-103.26\times21}{6}=\frac{3706 - 2168.46}{6}=\frac{1537.54}{6}\approx256.26$.
So the linear regression equation is $y=103.26x + 256.26$.

Step3: Determine the best - fit

To determine which equation best fits the data, we can look at the coefficient of determination ($R^{2}$ value). In general, the equation with a higher $R^{2}$ value is a better fit. However, without calculating the $R^{2}$ values, we can also observe the trend of the data. The data seems to have a non - linear growth pattern, and visually and conceptually, the exponential model seems to capture the trend better.

Answer:

(a) $y = 549.47\times1.04^{x}$
(b) $y=103.26x + 256.26$
(c) Exponential