Question

the table represents the results of survey that asked people their age and the percent of their budget that they spend on entertainment. age 27 34 68 25 41 39 52 70 50 % of budget for entertainment 10 7 5 9.5 6 9 4 2 6.8 write the function that best represents this data (do not enter spaces and round to nearest hundredths place). f(x)=

Explanation:

Step1: Assume linear regression model

We assume a linear - function of the form $y = ax + b$, where $x$ is age and $y$ is the percentage of budget for entertainment.

Step2: Calculate means

Let $x$ be the age values and $y$ be the percentage - of - budget values.
$\bar{x}=\frac{27 + 34+68+25+41+39+52+70+50}{9}=\frac{406}{9}\approx45.11$
$\bar{y}=\frac{10 + 7+5+9.5+6+9+4+2+6.8}{9}=\frac{59.3}{9}\approx6.59$

Step3: Calculate numerator and denominator for $a$

$n = 9$
$\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(27 - 45.11)(10 - 6.59)+(34 - 45.11)(7 - 6.59)+(68 - 45.11)(5 - 6.59)+(25 - 45.11)(9.5 - 6.59)+(41 - 45.11)(6 - 6.59)+(39 - 45.11)(9 - 6.59)+(52 - 45.11)(4 - 6.59)+(70 - 45.11)(2 - 6.59)+(50 - 45.11)(6.8 - 6.59)$
$=- 18.11\times3.41-11.11\times0.41 + 22.89\times(-1.59)-20.11\times2.91-4.11\times(-0.59)-6.11\times2.41+6.89\times(-2.59)+24.89\times(-4.59)+4.89\times0.21$
$=-61.7551-4.5551-36.3951 - 58.5201+2.4249-14.7251-17.8451-114.2451 + 1.0269$
$=-294.5988$
$\sum_{i = 1}^{n}(x_i-\bar{x})^2=(27 - 45.11)^2+(34 - 45.11)^2+(68 - 45.11)^2+(25 - 45.11)^2+(41 - 45.11)^2+(39 - 45.11)^2+(52 - 45.11)^2+(70 - 45.11)^2+(50 - 45.11)^2$
$=(-18.11)^2+(-11.11)^2+22.89^2+(-20.11)^2+(-4.11)^2+(-6.11)^2+6.89^2+24.89^2+4.89^2$
$=328.1721 + 123.4321+523.9521+404.4121+16.8921+37.3321+47.4721+619.5121+23.9121$
$=2124.091$
$a=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}=\frac{-294.5988}{2124.091}\approx - 0.14$

Step4: Calculate $b$

$b=\bar{y}-a\bar{x}=6.59-(-0.14)\times45.11=6.59 + 6.3154=12.9054\approx12.91$

Answer:

$f(x)=-0.14x + 12.91$