Meg plotted the graph below to show the relationship between the temperature of her city and the number of people at a swimming pool:
Main title on the graph is Swimming Pool Population. Graph shows 0 to 30 on x axis at increments of 5 and 0 to 12 on y axis at increments of 1. The label on the x axis is Temperature in degree C, and the label on the y axis is Number of People at the Pool. Dots are made at the ordered pairs 2.5, 1 and 5, 2 and 7.5, 2 and 7.5, 3 and 7.5, 4 and 10, 5 and 10, 6 and 12.5, 6 and 15, 7 and 15, 8 and 17.5, 5 and 17.5, 7 and 20, 9 and 22.5, 7 and 22.5, 9 and 25, 11 and 27.5, 12.
Part A: In your own words, describe the relationship between the temperature of the city and the number of people at the swimming pool. (5 points)
Part B: Describe how you can make the line of best fit. Write the approximate slope and y-intercept of the line of best fit. Show your work, including the points that you use to calculate slope and y-intercept. (5 points)
Part A: As the graph shows, there is a positive correlation between the temperature of the city and the number of people at the swimming pool. When the temperature increases, more people are likely to visit the swimming pool. This indicates that higher temperatures make it more desirable for people to go to the pool to cool off, while fewer people are likely to visit the pool on colder days.
Part B: To create a line of best fit, we can visually draw a straight line that passes through the middle of our data points. This line should have roughly an equal number of points above and below it. The line should be at such a position which minimizes the sum of the squares of the vertical distances between the line and each point.
Now, let's approximate the slope and y-intercept of the line of best fit.
First, we need to find two points on our line of best fit that we can use to calculate the slope. We can roughly approximate that the line of best fit goes through the points (5, 2) and (25, 11). These points may not exactly represent the actual line of best fit, but they are close enough for our purposes.
Next, we will calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Using our points (5, 2) and (25, 11):
m = (11 - 2) / (25 - 5)
m = 9 / 20
m = 0.45
Now, we can find the y-intercept (b) using the slope and one of our points. Let's use the point (5, 2):
y = mx + b
2 = 0.45(5) + b
2 = 2.25 + b
b = -0.25
So, the approximate equation for our line of best fit is:
y = 0.45x - 0.25