Question

question 10 (essay worth 10 points)
(04.05, 04.07 mc)
a student is looking at the relationship between the number of workers in a factory and the number of units produced daily. the table shows the data.

number of workers (x)	10	20	30	40	50	60	70	80	90
number of units (y)	60	81	159	202	256	310	352	395	450

part a: describe any correlation between the number of workers in a factory and the number of units produced daily. justify your answer. (4 points)
part b: write an equation for a line of fit or the line of best fit for the data. show all work. (3 points)
part c: what do the slope and y - intercept of the plot indicate about the scenario? (3 points)

Explanation:

Step1: Calculate the correlation

We can use a graphing utility or calculate the correlation coefficient $r$. As the number of workers ($x$) increases, the number of units ($y$) also generally increases, indicating a positive correlation. To justify, we can note that more workers mean more labor - power available to produce units.

Step2: Find the line of best - fit

We use the least - squares regression formula for a line $y = mx + b$, where $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\overline{y}-m\overline{x}$.
Let $n = 9$.
$\sum_{i = 1}^{9}x_i=10 + 20+\cdots+90=450$
$\sum_{i = 1}^{9}y_i=60 + 81+\cdots+450=1913$
$\sum_{i = 1}^{9}x_i^{2}=10^{2}+20^{2}+\cdots+90^{2}=28500$
$\sum_{i = 1}^{9}x_iy_i=10\times60 + 20\times81+\cdots+90\times450=122850$
$\overline{x}=\frac{\sum_{i = 1}^{9}x_i}{n}=\frac{450}{9}=50$
$\overline{y}=\frac{\sum_{i = 1}^{9}y_i}{n}=\frac{1913}{9}\approx212.56$
$m=\frac{9\times122850 - 450\times1913}{9\times28500-450^{2}}=\frac{1105650 - 860850}{256500 - 202500}=\frac{244800}{54000}=4.533$
$b = 212.56-4.533\times50=212.56 - 226.65=-14.09$
The line of best - fit is $y = 4.533x-14.09$.

Step3: Interpret the slope and y - intercept

The slope $m = 4.533$ indicates that for every additional worker, the number of units produced daily increases by approximately $4.533$ units. The y - intercept $b=-14.09$ has no real - world meaning in this context since having $x = 0$ workers and a negative number of units produced is not possible in the real world.

Answer:

Part A: There is a positive correlation as the number of units produced increases with the increase in the number of workers. Justification: More workers mean more labor available for production.
Part B: The line of best - fit is $y = 4.533x-14.09$.
Part C: The slope of approximately $4.533$ means that for each additional worker, about $4.533$ more units are produced daily. The y - intercept of $-14.09$ has no real - world significance as having a negative number of units when there are no workers is not possible.