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| 9 Theory slides |
| 7 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
The recommended reading is information that is helpful or necessary to understand before beginning the lesson.
Magdalena is fascinated by her local aquatic park and is eager to analyze how temperatures influence attendance. The following graph represents the data she collected — the average number of people that attend the park at specific temperatures.
Next week's forecast is extreme heat as temperatures will be around 110 degrees Farenheit. Using the forecast and her results, Magdalena wonders if she can predict the number of people who will visit the park. Can the park expect more than 10000 visitors?Valuable conclusions and predictions are made about a situation based on collected data. Before such statements can be made, the data is analyzed by using tools such as graphs. A scatter plot, for example, is used to identify the correlation between a pair of data sets.
A scatter plot is a graph that shows each observation of a bivariate data set as an ordered pair in a coordinate plane. Consider the following example, where a scatter plot illustrates the results gathered at a local ice cream parlor. This study records the number of ice creams sold and the corresponding air temperature.
Among other insights, the graph shows that when the temperature is about 100∘F, approximately 4000 ice creams are sold. Additionally, as the temperature increased, the number of sales also increased. In this case, it can be said that there is a positive correlation between the data sets — the number of ice creams sold and the air temperature.A correlation is a relation between two data sets. For example, consider two data sets, one consisting of temperatures and the other consisting of the number of coats sold. A decrease in the temperature may imply an increase in the number of coats sold. Based on the trend of the bivariate data, three types of correlations are possible which can be described using scatter plots.
Knowing the type of correlation helps analyze trends and make predictions based on data. Furthermore, the shape of the patterns formed by positive and negative correlations can be thought to have a positive and negative slope, respectively. The applet below shows how a data set transforms from a random pattern to a positive or a negative correlation.
The following applet shows different scatter plots. Select the type of correlation that matches the scatter plot shown.
Once the scatter plot of a data set is drawn and the type of correlation is identified, predictions can be made about the trend of the data by using lines of fit.
When data sets have a positive or negative correlation, the trend of the data can be modeled using a line of fit, also called a trend line. This line is drawn on a scatter plot near most of the data points, which appear evenly distributed above and below the line.
The scatter plot above shows the mean weights of kittens from the same litter in relation to their age. In this case, a line of fit could be drawn quite seamlessly. When drawing such a line of fit, the following characteristics should be considered.
At an aquatic park, a student-volunteer named Tadeo noticed a dedicated person who swims long distances in the lazy lagoon every Saturday morning.
Tadeo is amazed and wants to analyze how many calories the swimmer burns compared to the distance swam. He observes and records the swimmer diligently.
Distance (km) | Calories Burned |
---|---|
16 | 980 |
15 | 880 |
14 | 860 |
13 | 740 |
12 | 720 |
11 | 680 |
10 | 595 |
9 | 560 |
8 | 490 |
7 | 400 |
6 | 380 |
Zosia and Vincenzo are poster designers at the aquatic park. Right now, they are promoting a 3D movie about the life of dolphins called Above and Below the Line.
They recorded the number of tickets sold each week with the purpose of using the data to determine whether they should continue to advertise the movie on a billboard. The scatter plot shows the collected data.
Substitute (1,500) & (7,350)
Subtract terms
Put minus sign in front of fraction
Calculate quotient
x=1, y=500
Identity Property of Multiplication
LHS+25=RHS+25
Rearrange equation
In this lesson, it was taught how to analyze bivariate data using scatter plots and lines of fit. These mathematical concepts can now be used to solve the Challenge. It is now recognizable that Magdalena created a scatter plot to show the aquatic park visitors in relation to the temperatures.
Help Magdalena predict if more than 10000 people are expected to attend the park next week, given that the temperature will be around 110∘F. Justify the prediction.Yes, see solution.
The scatter plot shows that the number of attendants increases as the temperature increases, which means the data has a positive correlation. Therefore, it can be modeled with a line of fit.
By finding the equation of the line of fit, it can be predicted how many people would attend the park if the temperature is about 110∘F. To do so, the equation in slope-intercept form of a line can be used.
Consider the following table of values.
x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|---|---|---|
y | -20 | -12 | -11 | -8 | 0 | 3 | 5 | 13 | 13 | 21 | 25 |
Which scatter plot represents the data?
We will draw a scatter plot to determine which option represents the data. We will do it by treating the information in the table as a set of points.
We can see that this scatter plot corresponds to option A.
Consider the three general outcomes for the correlation of data.
Observation | Type of Correlation |
---|---|
As x increases, y increases. | Positive |
As x increases, y decreases. | Negative |
There is no visible pattern. | No |
Using this information, we can identify the correlation type of the given data. We can see that as the x-values increase, the y-values also increase.
To visualize it, let's draw a line of fit. The line of fit is not unique. However, ideally, the number of points below and above the line is expected to be the same.
We can conclude that the data has a positive correlation.
Determine if x and y show a positive correlation, a negative correlation, or no correlation.
We will analyze the scatter plot to determine the type of correlation the data shows. Let's first review the three general outcomes for the correlation of data.
Observation | Type of Correlation |
---|---|
As x increases, y increases. | Positive |
As x increases, y decreases. | Negative |
There is no visible pattern. | No Correlation |
We can now consider the scatter plot.
It appears that there is some kind of correlation. We will draw a line of fit to help us identify the type of correlation. Recall that the line of fit is not unique. However, the number of points below and above the line is expected to be similar.
Note that as the x-values increase, the y-values decrease. This means that there is a negative correlation between x and y.
Consider the given scatter plot.
Observing the graph, we see that the data does not follow any pattern. Therefore, there is no correlation between x and y.
Let's look at the given graph.
It seems that the data follows some pattern. We will draw a line of fit to identify the type of correlation the data follows.
We can see that as the x-values increase, the y-values also increase. This indicates that there is a positive correlation between x and y.
We will begin by looking at the given scatter plot.
Again, it appears that the data follows some pattern. A line of fit can help us to identify the type of correlation. Let's draw it!
We can note that as the x-values increase, the y-values decrease. This means there is a negative correlation between x and y.
Ali was reading a real estate magazine that he found in the garage of his house. He found out that the price of a house (in thousands of dollars) is connected to the number of square meters of the house. Two places called Ali's attention: Comfyville and Dreamville. The prices of the houses of each place are predicted by the following equations.
Price of Houses | |
---|---|
Comfyville | Dreamville |
p=159+0.05A | p=299+0.2A |
We are given two equations, and each can be used to predict the price p in thousands of dollars for a house in different cities given the area of the house in square feet, A.
Price of Houses | |
---|---|
Comfyville | Dreamville |
p=159+0.05 A | p=299+0.2 A |
The house Ali is looking at in the advertisement costs 933 thousands of dollars and has an area of 3200 square meters. Therefore, we need to evaluate both equations when A= 3200 and see which of the equations gives roughly 933.
Let's evaluate the equation for Comfyville.
Now, let's evaluate the equation for Dreamville.
We have evaluated both equations when A=3200. We can also note that none of the equations gave 933. However, because 939 is the closest to 933, the house shown in the advertisement is probably located in Dreamville.
The scatter plot shows the average number of people attending a beach club at specific temperatures.
To determine the average number of people attending the beach club when the temperature is 70^(∘)F, we will draw a horizontal segment to the y-axis to find its y-coordinate.
We can see the segment intercepts the y-axis at 100. This means that when it is 70^(∘)F, the average number of people attending the beach club is 100.
We now want to determine the average temperature given that the beach club had 350 visitors. In this case, we need to draw a vertical segment to the x-axis to determine the x-coordinate corresponding to the number of visitors.
Note that the segment intercepts the x-axis at 110. This means that the temperature last Saturday was 110^(∘)F.
To find which sentence best describes the scatter plot, let's first draw a line of fit through the points on the scatter plot.
Observing the line of fit, we can see that as the temperature increases the number of people attending the beach club also increases. This corresponds to option B.