Question

a small business tracks its total sales in the months after its grand opening. the data is shown in the table below. monthly sales month sales (in thousands) 1 15.2 2 17.5 3 19.0 4 22.1 5 24.8 using a linear line - of - best - fit model, approximately what are the predicted sales for month 8? 31600 31000 31700 32000

Explanation:

Step1: Calculate the slope ($m$)

Let $x$ be the month and $y$ be the sales (in thousands).
We use the formula $m=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}$, where $n = 5$.
$\sum x=1 + 2+3 + 4+5=15$, $\sum y=15.2+17.5 + 19.0+22.1+24.8 = 98.6$, $\sum xy=1\times15.2+2\times17.5+3\times19.0+4\times22.1+5\times24.8=15.2 + 35+57+88.4+124 = 319.6$, $\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=1 + 4+9+16+25 = 55$.
$m=\frac{5\times319.6-15\times98.6}{5\times55 - 15^{2}}=\frac{1598-1479}{275 - 225}=\frac{119}{50}=2.38$.

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

We use the formula $b=\frac{\sum y-m\sum x}{n}$.
$b=\frac{98.6-2.38\times15}{5}=\frac{98.6 - 35.7}{5}=\frac{62.9}{5}=12.58$.

Step3: Form the linear equation

The linear equation is $y = 2.38x+12.58$.

Step4: Predict for $x = 8$

Substitute $x = 8$ into the equation: $y=2.38\times8 + 12.58=19.04+12.58=31.62$.
Since $y$ is in thousands, the sales is $31.62\times1000 = 31620\approx31600$.

Answer:

31600