Question

this table shows the work experience and salaries of 4 accountants.
number of years experience (y) annual salary ($)
1 $53,000
4 $61,000
10 $75,000
12 $85,000
based on the information in the table, which linear equation best predicts salary, s, based on years of experience, y?
s = 2,762y + 49,857
s = 2,429y + 50,857
s = 2,667y + 50,333
s = 2,909y + 50,091

Explanation:

Step1: Calculate mean of y

$\bar{y} = \frac{1+4+10+12}{4} = \frac{27}{4} = 6.75$

Step2: Calculate mean of S

$\bar{S} = \frac{53000+61000+75000+85000}{4} = \frac{274000}{4} = 68500$

Step3: Compute slope (m)

First, calculate $\sum(y_i-\bar{y})(S_i-\bar{S})$ and $\sum(y_i-\bar{y})^2$:
$\sum(y_i-\bar{y})(S_i-\bar{S}) = (1-6.75)(53000-68500)+(4-6.75)(61000-68500)+(10-6.75)(75000-68500)+(12-6.75)(85000-68500)$
$= (-5.75)(-15500)+(-2.75)(-7500)+(3.25)(6500)+(5.25)(16500)$
$= 89125 + 20625 + 21125 + 86625 = 217500$

$\sum(y_i-\bar{y})^2 = (1-6.75)^2+(4-6.75)^2+(10-6.75)^2+(12-6.75)^2$
$= 33.0625 + 7.5625 + 10.5625 + 27.5625 = 78.75$

$m = \frac{217500}{78.75} = 2909.09 \approx 2909$

Step4: Compute intercept (b)

$b = \bar{S} - m\bar{y} = 68500 - 2909.09\times6.75 \approx 68500 - 18636 = 50091$

Step5: Form linear equation

$S = 2909y + 50091$

Answer:

D. $S = 2,909y + 50,091$