1. The table shows the number of soccer clubs and the population densities of different cities in a state.
City Pop. per square mile Clubs
1 160 6
2 173 8
3 254 7
4 284 11
5 315 9
6 322 10
7 459 14
8 487 20
Based on this information, which statement is true?
A. There is a strong correlation between two variables, and an increase in population density is likely associated with a larger number of soccer clubs.
B.There is a strong correlation between the two variables, and an increase in population density is unlikely to be associated with a larger number of soccer clubs.
C. There is a weak correlation between the two variables, but it is likely that an increase in population density is associated with a larger number of soccer clubs.
D. There is a weak correlation between the two variables, and it is unlikely that an increase in population density is associated with a larger number of soccer clubs.
2. Function Q and function R are both linear functions. Function Q has an x-intercept at (4,0), and the slope is 1/4. Function R is represented by the table.
x y
-4 14
3 0
5 -4
Which statement is true?
A. Function Q has a larger y-intercept and a less steep slope than function R
B. Function Q has a larger y-intercept and a steeper slope than function R
C. Function Q has a smaller y-intercept and a less steep slope than function R
D. Function Q has a smaller y-intercept and a steeper slope than function R
They kinda squished together, but the first list of numbers is City: 1 2 3 4 5 6 7 8 Population per square mile: 160 173 254 284 315 322 459 487 Clubs: 6 8 7 11 9 10 14 20
divide?
1. To determine the correlation between population density and the number of soccer clubs, we can examine the data in the table. The data shows that as the population density increases, the number of soccer clubs generally increases as well:
City Pop. per square mile Clubs
1 160 6
2 173 8
3 254 7
4 284 11
5 315 9
6 322 10
7 459 14
8 487 20
From the data, we can see that as the population density increases from city 1 to city 8, the number of soccer clubs also increases. Therefore, we can conclude that there is a correlation between population density and the number of soccer clubs.
To determine the strength of the correlation, we need to consider the variability in the data and how closely the points follow a linear trend. Based on the data provided, it appears that there is a fairly strong correlation between population density and the number of soccer clubs. The points on the graph are not scattered randomly; instead, they generally follow a positive linear trend.
Therefore, the correct statement is:
A. There is a strong correlation between two variables, and an increase in population density is likely associated with a larger number of soccer clubs.
2. To compare functions Q and R, let's examine the given information:
Function Q has an x-intercept at (4,0) and a slope of 1/4. This means that the function crosses the x-axis at x = 4 and has a slope of 1/4, which represents a moderate increase in the y-coordinate for each unit increase in the x-coordinate.
Function R is represented by the table:
x y
-4 14
3 0
5 -4
Based on the table, we can plot the points (-4, 14), (3, 0), and (5, -4) on a graph. This will help us determine the slope and y-intercept of function R.
The slope of a linear function can be calculated by finding the change in y-coordinates divided by the change in x-coordinates:
slope = (change in y) / (change in x)
Using the points (3, 0) and (-4, 14):
(change in y) = 0 - 14 = -14
(change in x) = 3 - (-4) = 7
slope = (-14) / 7 = -2
From the data provided, the slope of function R is -2.
To determine the y-intercept of function R, we can substitute one of the points into the equation y = mx + b, where m is the slope and b is the y-intercept. Let's use the point (3, 0):
0 = (-2)(3) + b
0 = -6 + b
b = 6
So, the y-intercept of function R is 6.
Comparing the characteristics of functions Q and R:
Function Q has a smaller y-intercept (0) compared to function R (6), and a steeper slope (1/4) compared to function R (-2).
Therefore, the correct statement is:
D. Function Q has a smaller y-intercept and a steeper slope than function R.
To answer these questions, let's analyze the given information and apply relevant mathematical concepts.
1. For the first question, we are given a table with two variables: population density and number of soccer clubs in different cities. We need to determine the relationship between these variables.
To identify the correlation between the variables, we can calculate the correlation coefficient. However, since we don't have the complete data, calculating the correlation coefficient is not possible. Instead, we can examine the patterns in the data.
Looking at the table, we see that as the population density increases, the number of soccer clubs generally tends to increase. City 1 has the smallest population density and the lowest number of soccer clubs, while City 8 has the highest population density and the highest number of soccer clubs.
Based on this pattern, we can conclude that there is likely an association between population density and the number of soccer clubs. Therefore, the answer is:
C. There is a weak correlation between the two variables, but it is likely that an increase in population density is associated with a larger number of soccer clubs.
2. For the second question, we are given information about two linear functions, Q and R. We need to compare their y-intercepts and slopes based on the given information.
Function Q has an x-intercept at (4,0), which means the graph of Q crosses the x-axis at x = 4 and y = 0. Additionally, the slope of Q is 1/4.
Function R is represented by a table of values. We can use the given information to find the slope and y-intercept of R.
Using the two points (-4, 14) and (3, 0), we can calculate the slope of R:
Slope (m) = (change in y) / (change in x) = (0 - 14) / (3 - (-4)) = -14 / 7 = -2
We can also find the y-intercept of R by substituting the values of any point (e.g., (3, 0)) and the slope into the equation y = mx + b and solving for b:
0 = (-2)(3) + b
0 = -6 + b
b = 6
By comparing the given information, we can conclude that Function Q has a smaller y-intercept and a steeper slope than Function R. Therefore, the answer is:
D. Function Q has a smaller y-intercept and a steeper slope than function R.