Question

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: Descnbe any correlation between the number of workers in a factory and the number of units produced daily. Justify your answer. (4 points)

Part B: Wite an equation tor a line of it or the line or best itorine data. show all work points)

Part C: What do the slope and y intercept of the plot indicate about the scenano? (3 points)

Answer 1

As the number of workers increases, the number of units produced daily also increases. The denominator is zero, the slope (m) is undefined. This could indicate that the relationship is not strictly linear, or there may be an error in the data.

Part A: Describing the Correlation

To determine the correlation between the number of workers (x) and the number of units produced daily (y), we can examine the trend in the data.

As the number of workers increases, the number of units produced daily also increases. This suggests a positive correlation between the two variables.

In other words, there is a tendency for an increase in the number of workers to be associated with an increase in the number of units produced.

Part B: Equation for the Line of Best Fit

To find the equation for the line of best fit, we can use linear regression. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

The formula for the slope (m) can be calculated as follows:

`\[ m = (n(\sum xy) - (\sum x)(\sum y))/(n(\sum x^2) - (\sum x)^2) \]`

The formula for the y-intercept (b) is:

`\[ b = ((\sum y) - m(\sum x))/(n) \]`

where
`\( n \)` is the number of data points,
`\( \sum xy \)` is the sum of the product of x and y,
`\( \sum x \)` is the sum of x values,
`\( \sum y \)` is the sum of y values, and
`\( \sum x^2 \)` is the sum of the squares of x values.

Calculating the values:

`\[ n = 9, \sum x = 540, \sum y = 2561, \sum xy = 201189, \sum x^2 = 32400 \]`

`\[ m = ((9 * 201189) - (540 * 2561))/((9 * 32400) - 540^2) \]`

`\[ m = (1810701 - 1386540)/(291600 - 291600) \]`

`\[ m = (423161)/(0) \]`

Since the denominator is zero, the slope (m) is undefined. This could indicate that the relationship is not strictly linear, or there may be an error in the data.

Part C: Interpretation of the Slope and Y-Intercept

In a linear equation of the form
`\( y = mx + b \)`, the slope (m) represents the rate of change of y with respect to x. In this context, if the slope were defined, it would indicate how much the number of units produced daily changes for each additional worker.

The y-intercept (b) represents the value of y when x is zero. In the context of this scenario, it would be the estimated number of units produced when there are no workers. As this cannot be possible, the given data cannot indicate about the scenario