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This lesson will explore how two quantities are related and how to make predictions by finding a line of fit for scatter plots.
### Catch-Up and Review

**The recommended reading is information that is helpful or necessary to understand before beginning the lesson.**

- These are some topics that help in data analysis.
- These are some important topics about linear equations.

The following applet shows three graphs — each with a set of points distributed on a coordinate plane.

Pay close attention to the behavior of the dependent variable as the independent variable increases. Does $y$ increase, decrease, or can it not be distinguished?

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 in which a scatter plot shows the results obtained at a local ice cream parlor of a study that recorded the number of ice creams sold and the corresponding air temperature.

Among other insights, the graph shows that when the temperature was about $100_{∘}F,$ approximately $4000$ ice creams were sold. Additionally, as the temperature increased, the number of sales increased. In this case, it can be said that there is a positive correlation between the data sets — number of ice creams sold and 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.

- The data needs to have either a positive or negative correlation.
- While a line of fit is not unique and does not create an exact distribution, ideally, about half of the points should be above the line and about half below the line.
- An equation of the line can be found using two of its points. These points do not necessarily belong to the bivariate data set.

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 |
---|---|

$16$ | $980$ |

$15$ | $880$ |

$14$ | $860$ |

$13$ | $740$ |

$12$ | $720$ |

$11$ | $680$ |

$10$ | $595$ |

$9$ | $560$ |

$8$ | $490$ |

$7$ | $400$ |

$6$ | $380$ |

a Make a scatter plot of the data.

b What type of correlation does the data have? Justify the answer.

c Draw a line of fit for the scatter plot.

d Find an equation for the line of fit.

a **Example Answer: **

b Positive, see solution.

c **Example Answer: **

d **Example Equation: ** $y=50x+110$

a To draw the scatter plot, let $x$ be the distance in kilometers and $y$ the calories burned. With this in mind, the information from the table can be shown on a scatter plot.

b From the scatter plot previously drawn, it can be seen that as the distance increases, the number of calories burned also increases. Therefore, the bivariate data has a positive correlation.

c Since the data has a positive correlation, it can be modeled with 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 similar.

d Because the equation of a line can be found using any two points on the line, two points whose coordinates can be easily identified will be marked on the graph of the line of fit.

$y−y_{1}y−y_{1} =m(x−x_{1})⇓=50(x−x_{1}) $

To complete the equation, any of the two points can be substituted above. For simplicity, $(4,310)$ will be used.
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.

a Draw a line of fit for the scatter plot.

b Find an equation for the line of fit in slope-intercept form.

c If the *expected* number of tickets sold on week $12$ is more than $150000,$ Vincenzo and Zosia will keep the movie for at least two more weeks on the billboard. Use the line of fit to predict if the movie stays on the billboard.

a **Example Line: **

b **Example Equation: ** $y=-25x+525$

c **Example Answer: ** Because the expected number of tickets sold on week $12$ is about $237000,$ they may decide to keep the movie.

a What type of correlation does the scatter plot have?

b Use two points on the line of fit to find the slope and the $y-$intercept of the line.

c Evaluate the equation found in Part B for $x=12.$

a The scatter plot shows that the number of tickets sold decreases as time passes. Therefore, the data has a negative correlation and can be modeled by a line of fit. Recall that the line of fit is close to most of the data points, while the points are ideally half above and half below the line of fit.

b The equation of a line in slope-intercept form has the following form.

$y=mx+b $

In this equation, $m$ is the slope and $b$ the $y-$intercept of the line. The slope of the line of fit can be found by using the Slope Formula.
$m=x_{2}−x_{1}y_{2}−y_{1} $

The points $(1,500)$ and $(7,350)$ are on the line of fit. This means that they can be substituted in the above formula.
$m=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(1,500)$ & $(7,350)$

$m=7−1350−500 $

Evaluate right-hand side

SubTerms

Subtract terms

$m=6-150 $

MoveNegNumToFrac

Put minus sign in front of fraction

$m=-6150 $

CalcQuot

Calculate quotient

$m=-25$

$y=mx+bsubstitute y=-25x+b $

Now, the $y-$intercept can be found by substituting any of the points into the partial equation. In this case, $(1,500)$ will be used. $y=-25x+b$

SubstituteII

$x=1$, $y=500$

$500=-25(1)+b$

Solve for $b$

IdPropMult

Identity Property of Multiplication

$500=-25+b$

AddEqn

$LHS+25=RHS+25$

$525=b$

RearrangeEqn

Rearrange equation

$b=525$

$y=-25x+bsubstitute y=-25x+525 $

c The line of fit describes the trend of the data. This means that it can be used to predict how many tickets would be sold on the $12_{th}$ week. Therefore, by evaluating the equation of the line of fit for $x=12,$ the expected number of tickets sold can be found.

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.$y=mx+b $

In this equation, $m$ is the slope and $b$ the $y-$intercept of the line. The slope can be calculated by using the Slope Formula.
$m=x_{2}−x_{1}y_{2}−y_{1} $

Because the points $(70,7000)$ and $(80,8000)$ are on the line of fit, they will be used to find the slope.
$m=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(70,7000)$ & $(80,8000)$

$m=80−708000−7000 $

$m=100$

$y=mx+bsubstitute y=100x+b $

By substituting one of the points on the line of fit, the $y-$intercept can be found. In this case, $(70,7000)$ will be used.
Therefore, the $y-$intercept $b$ is $0.$ With this information, the equation of the line of fit can be written.
$y=100x+bsubstitute yy =100x+0=100x $

Finally, using this equation, the number of people that would attend the park if the temperature is about $110_{∘}F$ can be predicted. To do so, the equation needs to be evaluated for $x=110.$
It is expected that more than $10000$ people will attend the park next week. What a brilliant way to make a prediction. Magdalena has really helped the aquatic park prepare for the influx of visitors surely to come.