Reach Out<\/strong><\/p>\nReach for the skies! Who in your class can reach the highest? That height depends on how tall each person is and the lengths of their arms.
\nNow stick your arms out to your sides. Sometimes this horizontal distance is called your wingspan. The wandering albatross can have a wingspan of up to 12 feet. How does your wingspan compare? Think about this during the 3-Act Mathematical Modeling lesson.
\n<\/p>\n
enVision STEM Project<\/strong><\/p>\nDid You Know?<\/strong>
\nA fishery biologist collects data on fish, such as the size and health of the fish population in a particular body of water.
\n
\nLargemouth bass and smallmouth bass are the most popular game fish in North America.
\n
\nBiologists often use tagging studies to estimate fish population, as well as to estimate catch and harvest rates.
\n
\nThe average lifespan of bass is about 16 years, but some have lived more than 20 years.<\/p>\nResearch suggests that bass can see red better than any other color on the spectrum.
\n<\/p>\n
Your Task: How Many Fish?<\/strong><\/p>\nSuppose a fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged. How many basses are in the lake? You and your classmates will explore how the biologist can use sampling to describe patterns and to make generalizations about the entire population.
\n
\nAnswer:
\nIt is given that
\nA fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged.
\nSo,
\nThe total number of basses = 500 + 200
\n= 700
\nThe number of basses tagged = 30
\nSo,
\nAfter netting, the number of basses = 700 – 30
\n= 670
\nHence, from the above,
\nWe can conclude that after the biologist nets 200 basses,
\nThe total number of basses present are: 670<\/p>\n
<\/p>\n
Topic 4 GET READY!<\/h3>\n
Review What You Know!<\/strong><\/p>\nVocabulary<\/strong><\/p>\nChoose the best term from the box to complete each definition.<\/p>\n
Question 1.
\n____ is the change in y divided by the change in x.
\nAnswer:
\nWe know that,
\n“Slope” is the change in y divided by the change in x
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Slope<\/p>\n
Question 2.
\nA relationship where for every x units of one quantity there are y units of another quantity is a ____
\nAnswer:
\nWe know that,
\nA relationship where for every x units of one quantity, there are y units of another quantity is a “Ratio”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Ratio<\/p>\n
Question 3.
\nThe ____ is the horizontal line in a coordinate plane.
\nAnswer:
\nWe know that,
\nThe “X-axis” is the horizontal line in a coordinate plane
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: X-axis<\/p>\n
Question 4.
\nThe ___ is the vertical line in a coordinate plane.
\nAnswer:
\nWe know that,
\nThe “Y-axis” is the vertical line in a coordinate plane
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Y-axis<\/p>\n
Graphing Points<\/strong>
\nGraph and label each point on the coordinate plane.<\/p>\n<\/p>\n
Question 5.
\n(-2, 4)
\nAnswer:
\n<\/p>\n
Question 6.
\n(0, 3)
\nAnswer:
\n<\/p>\n
Question 7.
\n(3, -1)
\nAnswer:
\n<\/p>\n
Question 8.
\n(-4, -3)
\nAnswer:
\n<\/p>\n
Finding Slope<\/strong><\/p>\nFind the slope between each pair of points.<\/p>\n
Question 9.
\n(4, 6) and (-2, 8)
\nAnswer:
\nThe given points are: (4, 6), (-2, 8)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{8 – 6}{-2 – 4}\\)
\n= \\(\\frac{2}{-6}\\)
\n= –\\(\\frac{1}{3}\\)
\nHence, from the above,
\nWe can conclude that the slope between the given points is: –\\(\\frac{1}{3}\\)<\/p>\nQuestion 10.
\n(-1, 3) and (5,9)
\nAnswer:
\nThe given points are: (-1, 3), (5, 9)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{9 – 3}{5 + 1}\\)
\n= \\(\\frac{6}{6}\\)
\n= 1
\nHence, from the above,
\nWe can conclude that the slope between the given points is: 1<\/p>\nQuestion 11.
\n(5, -1) and (-3, -7)
\nAnswer:
\nThe given points are: (5, -1), (-3, -7)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{-7 + 1}{-3 – 5}\\)
\n= \\(\\frac{-6}{-8}\\)
\n= \\(\\frac{3}{4}\\)
\nHence, from the above,
\nWe can conclude that the slope between the given points is: \\(\\frac{3}{4}\\)<\/p>\nWriting Fractions as Percents<\/p>\n
Question 12.
\nExplain how to write \\(\\frac{36}{60}\\) as a percent.
\nAnswer:
\nThe given fraction is: \\(\\frac{36}{60}\\)
\nWe know that,
\n1 = 100%
\nSo,
\n\\(\\frac{36}{60}\\)
\n= \\(\\frac{36}{60}\\) \u00d7 100%
\n= \\(\\frac{36 \u00d7 100%}{60}\\)
\n= 60%
\nHence, from the above,
\nWe can conclude that the value of \\(\\frac{36}{60}\\) as a percent is: 60%<\/p>\n
<\/p>\n
Language Development<\/strong><\/p>\nComplete the graphic organizer. Write the definitions of the terms in your own words. Use words or a sketch to show an example.<\/p>\n
\n
\n
\nAnswer:
\n<\/p>\n
Topic 4 PICK A PROJECT<\/h3>\n
PROJECT 4A<\/strong><\/p>\nWhat carnival games do you have a good chance of winning, and why?
\nPROJECT: BUILD A CARNIVAL GAME
\n<\/p>\n
PROJECT 4B<\/strong>
\nIf you had a superpower, what would it be?
\nPROJECT: SUMMARIZE SUPERHERO DATA
\n<\/p>\nPROJECT 4C<\/strong>
\nWhat makes a song’s lyrics catchy?
\nPROJECT: WRITE A SONG
\n<\/p>\nPROJECT 4D<\/strong>
\nHow does your dream job use math?
\nPROJECT: RESEARCH A CAREER
\n<\/p>\nLesson 4.1 Construct and Interpret Scatter Plots<\/h3>\n
Solve & Discuss It!<\/strong><\/p>\nLuciana is starting a two-week social media campaign to attract new subscribers to Blaston, a music website for teens. She has the following data from her last campaign to help plan her strategy.<\/p>\n
<\/p>\n
Look for Relationships<\/strong>
\nHow are the number of media posts and the number of subscribers related?
\nAnswer:
\nThe given data is:
\n
\nIf we observe the total data,
\nWe can find that there is no particular pattern for the given data
\nBut, if we observe the given data in parts, then
\nFrom 1 – 5 posts per day,
\nSocial media posts per day \u221d New subscribers
\nFrom 8 – 10 posts per day,
\nSocial media posts per day \u221d \\(\\frac{1}{New subscribers}\\)
\nFrom 6 – 7 posts per day,
\nThere is no pattern between the social media posts per day and the number of subscribers<\/p>\nBased on this data, what should be Luciana’s strategy for the new campaign?
\nAnswer:
\nBased on the pattern of the given data (As mentioned above),
\nLuciana’s strategy for the new campaign must be:
\nThe social media posts per day and the new subscribers must be in an increasing trend only<\/p>\n
<\/p>\n
Focus on math practices<\/strong>
\nUse Structure What patterns do you see in the data from Luciana’s last social media campaign?
\nAnswer:
\nIf we observe the total data,
\nWe can find that there is no particular pattern for the given data
\nBut, if we observe the given data in parts, then
\nFrom 1 – 5 posts per day,
\nSocial media posts per day \u221d New subscribers
\nFrom 8 – 10 posts per day,
\nSocial media posts per day \u221d \\(\\frac{1}{New subscribers}\\)
\nFrom 6 – 7 posts per day,
\nThere is no pattern between the social media posts per day and the number of subscribers<\/p>\nEssential Question<\/strong>
\nHow does a scatter plot show the relationship between paired data?
\nAnswer:
\nThe\u00a0scatter\u00a0diagram graphs\u00a0pairs of\u00a0numerical\u00a0data, with one variable on each axis,\u00a0to\u00a0look for a\u00a0relationship between\u00a0them. If the variables\u00a0are\u00a0correlated, the points will fall along a line or curve. The better the\u00a0correlation, the tighter the points will hug the line.<\/p>\nTry It!<\/strong>
\nLuciana collects data about the number of entries and the ages of the subscribers who enter the concert giveaway.<\/p>\n<\/p>\n
The point that represents the data in the fourth column has coordinates
\nAnswer:
\nThe given data is:
\n
\nNow,
\nWe know that,
\nThe x-axis always represents the independent variables
\nThe y-axis always represents the dependent variables
\nSo,
\nFrom the given data,
\nThe independent variable (x) is: Age
\nThe dependent variable (y) is: Number of entries
\nWe know that,
\nThe ordered pair must be in the form of (x, y)
\nHence, from the above,
\nWe can conclude that the point that represents the data in the fourth column has coordinates (13, 9)<\/p>\n
Convince Me!<\/strong>
\nExplain how Luciana would choose scales for the x-axis and y-axis.
\nAnswer:
\nIn a graph,
\nThe scales for the x-axis and the y-axis is nothing but the rate of change between the values of x and y
\nEx:
\nThe given data is:
\n
\nFrom the above data,
\nThe scale for the x-axis is:
\nRate of change between the values of x = 11 – 10 (or) 12 – 11 and so on
\n= 1
\nThe scale for the y-axis is:
\nRate of change between the values of y = 9 – 8 (or) 10 – 9
\n= 1
\nHence, from the above,
\nWe can conclude that
\nThe scale for the x-axis is:
\n1 unit = 1 year
\nThe scale for the y-axis is:
\n1 unit = 1 Entry<\/p>\nTry It!<\/strong>
\nAvery also tracks the number of minutes a player plays and the number of points the player scored. Describe the association between the two data sets. Tell what the association suggests.
\n
\nAnswer:
\nIt is given that
\nAvery also tracks the number of minutes a player plays and the number of points the player scored.
\nSo,
\nThe given scatter plot is:
\n
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the overall pattern is in an increasing trend
\nBut,
\nWhen we observe the number of points in the perspective of minutes,
\nThe pattern is in clusters
\nNow,
\nAt 4 minutes,
\nThe number of points scored is: 0
\nAt 6 and 8 minutes,
\nThe number of points scored is: 2
\nBetween 8 and 14 minutes,
\nThe number of points scored is: 4
\nBetween 10 and 16 minutes,
\nThe number of points scored is: 6<\/p>\nKEY CONCEPT<\/strong><\/p>\nA scatter plot shows the relationship, or association, between two sets of data.
\n
\nThe y-values increase as the x-values increase.
\n
\nThe y-values decrease as the x-values increase.
\n
\nThere is no consistent pattern between the y-values and the x-values.<\/p>\n
Do You Understand?<\/strong><\/p>\nQuestion 1.
\n? Essential Question<\/strong>
\nHow does a scatter plot show the relationship between paired data?
\nAnswer:
\nThe\u00a0scatter\u00a0diagram graphs\u00a0pairs of\u00a0numerical\u00a0data, with one variable on each axis,\u00a0to\u00a0look for a\u00a0relationship between\u00a0them. If the variables\u00a0are\u00a0correlated, the points will fall along a line or curve. The better the\u00a0correlation, the tighter the points will hug the line.<\/p>\nQuestion 2.
\nModel with Math<\/strong>
\nMarcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores. What scales for the axes might be best for constructing the scatter plot?
\nAnswer:
\nIt is given that
\nMarcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores.
\nSo,
\nThe scales for the axes that might be best is:
\nFor the x-axis:
\nIt is given that Macy always sleeps fewer than 9 hours
\nSo,
\nThe scale might be: 1 unit starting from 9 to 24
\nFor the y-axis:
\nIt is given that Macy never scored more than 27 points in a basketball game
\nSo,
\nThe scale might be: 1 unit starting from 27 to the corresponding last value of x
\nHence, from the above,
\nWe can conclude that
\nFor the x-axis,
\nThe scale is:
\n1 unit = 1 hour
\nFor the y-axis,
\nThe scale is:
\n1 unit = 1 point<\/p>\nQuestion 3.
\nConstruct Arguments<\/strong> Kyle says that every scatter plot will have a cluster, gap, and outlier. Is he correct? Explain.
\nAnswer:
\nWe know that,
\nA scatter plot might have a cluster, a gap, and an outlier or the association of any two but not all three are present
\nHence, from the above,
\nWe can conclude that Kyle is not correct<\/p>\nDo You Know How?<\/strong><\/p>\nQuestion 4.
\nPhoebe constructs a scatter plot to show the data. What scales could she use for the x- and y-axes?
\n
\nAnswer:
\nIt is given that
\nPhoebe constructs a scatter plot to show the data.
\nNote:
\nThe scatter plot is drawn only for the relations that are functions and we know that the rate of change is constant for a function
\nNow,
\nThe given data is:
\n
\nFrom the given data,
\nThe scale she could use for the x-axis will be 1 unit
\nThe scale she might use for the y-axis will be: 1 unit
\nHence, from the above,
\nWe can conclude that
\nThe scale used for the x-axis is:
\n1 unit = 1 inch (Shoe size)
\nThe scale used for the y-axis is:
\n1 unit = 1 inch (Height)<\/p>\n
Question 5.
\nGermaine constructs a scatter plot to show how many people visit different theme parks in a month. Why might clusters and outliers be present?
\n
\nAnswer:
\nIt is given that
\nGermaine constructs a scatter plot to show how many people visit different theme parks in a month
\nNow,
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that the graph is non-linear
\nWe know that,
\nA\u00a0cluster\u00a0is a group of objects, numbers, data points (information), or even people that are located close together
\nAn\u00a0outlier\u00a0is a value in a data set that is very different from the other values. That is,\u00a0outliers\u00a0are values unusually far from the middle
\nSo,
\nFrom the given scatter plot,
\nWe can observe that there are 2 points that are far from the pattern and those points are called “Outliers”
\nSince the scatter plot is non-linear, the points will be grouped together and the group of points is called “Clusters”<\/p>\n
Practice & Problem Solving<\/strong><\/p>\nQuestion 6.
\nLeveled Practice<\/strong> The table shows the racing times in minutes for the first two laps in a race. Complete the scatter plot.
\n
\nAnswer:
\nIt is given that
\nThe table shows the racing times in minutes for the first two laps in a race
\nNow,
\nIn the graph,
\nThe x-axis represents: The racing times of Lap 1
\nThe y-axis represents: The racing times of Lap 2
\nHence,
\nThe completed scatter plot with the x-axis and y-axis is:
\n<\/p>\nQuestion 7.
\nThe scatter plot represents the prices and number of books sold in a bookstore.
\na. Identify the cluster in the scatter plot and explain what it means.
\n
\nAnswer:
\nIt is given that
\nThe scatter plot represents the prices and number of books sold in a bookstore.
\nNow,
\nThe given scatter plot is:
\n
\nNow,
\nWe know that,
\nA\u00a0cluster\u00a0is a group of objects, numbers, data points (information), or even people that are located close together
\nSo,
\nFrom the above scatter plot,
\nWe can observe that the clusters are present between the intervals of 10 books sold and 20 books sold<\/p>\n
b. Generalize<\/strong> How does the scatter plot show the relationship between the data points? Explain.
\nAnswer:
\nThe\u00a0<\/span>scatter\u00a0diagram graphs\u00a0<\/span>pairs of\u00a0numerical\u00a0<\/span>data, with one variable on each axis,\u00a0<\/span>to\u00a0look for a\u00a0<\/span>relationship between\u00a0them. If the variables\u00a0<\/span>are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.<\/span><\/p>\nQuestion 8.
\nThe table shows the monthly attendance in thousands at museums in one country over a 12-month period.
\n
\na. Complete the scatter plot to represent the data.
\nAnswer:
\nIt is given that
\nThe table shows the monthly attendance in thousands at museums in one country over a 12-month period.
\nNow,
\nThe given data is:
\n
\nHence,
\nThe representation of the scatter plot for the given data is:
\n<\/p>\n
b. Identify any outliers in the scatter plot.
\nAnswer:
\nWe know that,
\nA value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
\nHence, from the above,
\nWe can conclude that
\nThe outliers in the scatterplot are at (12, 3), (6, 36)<\/p>\n
c. What situation might have caused an outlier?
\n
\nAnswer:
\nFrom part (b),
\nThe outliers in the scatterplot are at (12, 3), (6, 36)
\nSo,
\nFrom the given points,
\nWe can conclude that the number of people is very low at that particular month to cause the situation of outliers<\/p>\n
Question 9.
\nHigher-Order Thinking<\/strong> The table shows the number of painters and sculptors enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data.
\n
\na. What error did Jashar likely make?
\n
\nAnswer:
\nIt is given that
\nThe table shows the number of painters and sculptors enrolled in seven art schools.
\nNow,
\nThe given data is:
\n
\nSo,
\nFrom the given data,
\nWe can observe that
\nThe variable that will be on the x-axis (Independent variable) is: Number of painters
\nThe variable that will be on the y-axis (Dependent variable) is: Number of sculptors
\nBut,
\nFrom the scatter plot drawn by Jashar,
\nHe interchanged the variables of the axes i.e., he took the independent variable at the y-axis and the dependent variable at the x-axis
\nHence, from the above,
\nWe can conclude that the error made by Jashar is the interchange of the variables of the axes<\/p>\nb. Explain the relationship between the number of painters and sculptors enrolled in the art schools.
\n
\nAnswer:
\nFrom the given data,
\nWe can observe that for the increase in painters, the number of sculptors also increases
\nHence, from the above,
\nWe can conclude that the relationship between the number of painters and sculptors enrolled in the art schools is:
\nNumber of painters \u221d Number of sculptors<\/p>\n
c. Reasoning<\/strong> Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.
\nAnswer:
\nWe know that,
\nA value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
\nNow,
\nThe scatter plot for the given data is:
\n
\nHence, from the above,
\nWe can conclude that the two possible outliers identified by Jashar are: (11, 6), and (20, 45)<\/p>\nAssessment Practice<\/strong><\/p>\nUse the scatter plot to answer 10 and 11.<\/p>\n
Question 10.
\nTen athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times. Select all statements that are true.
\n
\n Nine of the times for the first race were at least 16 seconds.
\n Eight of the times for the second race were less than 17 seconds.
\n There were seven athletes who were faster in the second race than in the first.
\n There were three athletes who had the same time in both races.
\n There were three athletes whose times in the two races differed by exactly 1 second.
\nAnswer:
\nIt is given that
\nTen athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times.
\nNow,
\nThe given scatter plot is:
\n
\nHence,
\nThe correct statements about the given scatter plot is:<\/p>\n
Question 11.
\nWhat was the greatest difference for a single runner in finishing times in the races?
\nA. 3 seconds
\nB. 4 seconds
\nC. 5 seconds
\nD. 7 seconds
\nAnswer:
\nFrom the given scatter plot,
\nWe can observe that
\nThe lowest time a runner takes for completing a race is approximately 14 seconds
\nThe highest time a runner takes for completing a race is approximately 17 seconds
\nSo,
\nThe greatest difference for a single runner in finishing times in the races is: 3 seconds
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n
Lesson 4.2 Analyze Linear Associations<\/h3>\n
Solve & Discuss It!<\/strong><\/p>\nAngus has a big test coming up. Should he stay up and study or go to bed early the night before the test? Defend your recommendation.
\n
\nAnswer:
\nIt is given that
\nAngus has a big test coming up
\nNow,
\nThe relationship between the sleeping time and the percentage of marks is also given
\nNow,
\nFrom the given data,
\nWe can observe that
\nIf he went to bed early i.e., at 9:00, then he got 93%
\nIf he studied until 11:00, then he got only 92%
\nHence, from the above,
\nWe can conclude that Angus has to go to bed early before the big test<\/p>\n
Generalize
\n<\/strong>Can you make a general statement about which option leads to a better result?
\nAnswer:
\nGenerally, going to bed early will lead to better results before a test<\/p>\nFocus on math practices<\/strong>
\nConstruct Arguments<\/strong> What other factors should Angus also take into consideration to make a decision? Defend your response.
\nAnswer:
\nThe other factors that Angus should also take into consideration when making a decision are:
\nA) Nature of the exam
\nB) Coverage of the topics for the particular exam
\nC) Number of revisions<\/p>\n? Essential Question<\/strong>
\nHow can you describe the association of two data sets?
\nAnswer:
\nAssociation (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none<\/p>\nTry It!<\/strong>
\nGeorgia and her classmates also measured their foot length. Use a pencil to find the trend line. Sketch the trend line for the scatter plot.
\n
\nAnswer:
\nIt is given that
\nGeorgia and her classmates also measured their foot length
\nHence,
\nThe representation of a trend line for the given scatter plot is:
\n<\/p>\nTry It!<\/strong>
\nFor each scatter plot, identify the association between the data. If there is no association, state so.
\na.
\n<\/p>\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the above scatter plot,
\nWe can observe that the points are all scattered
\nHence, from the above,
\nWe can conclude that the given scatter plot has a weaker association<\/p>\n
b.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nThe points are all in a non-linear shape
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association
\nc.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the scatter plot,
\nWe can observe that the points are all scattered
\nHence, from the above,
\nWe can conclude that the given scatter plot has a weaker association<\/p>\n
KEY CONCEPT<\/strong><\/p>\nScatter plots can show a linear association, a nonlinear association, or no association. For scatter plots that suggest a linear association, you can draw a trend line to show the association. You can assess the strength of the association by looking at the distances of plotted points from the trend line.
\n<\/p>\n
Do You Understand?<\/strong><\/p>\nQuestion 1.
\n? Essential Question<\/strong> How can you describe the relationship between the two sets of data?
\nAnswer:
\nAssociation (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none<\/p>\nQuestion 2.
\nLook for Relationships<\/strong> How does a trend line describe the strength of the association?
\nAnswer:
\nThe straight\u00a0<\/span>line\u00a0is a\u00a0<\/span>trend line, designed to come as close as possible to all the data points. The\u00a0<\/span>trend line\u00a0has a positive slope, which shows a positive\u00a0<\/span>relationship\u00a0between X and Y. The points in the\u00a0<\/span>graph\u00a0are tightly clustered about the\u00a0<\/span>trend line\u00a0due to the strength of the\u00a0<\/span>relationship between X and Y.<\/span><\/p>\nQuestion 3.
\nConstruct Arguments<\/strong> How does the scatter plot of a nonlinear association differ from that of a linear association?
\nAnswer:
\nScatterplots with a\u00a0linear\u00a0pattern have points that seem to generally fall along a line while\u00a0nonlinear\u00a0patterns seem to follow along some curve. Whatever the pattern is, we use this to describe the\u00a0association\u00a0between the variables.<\/p>\nDo You Know How?<\/strong><\/p>\nQuestion 4.
\nDescribe the association between the two sets of data in the scatter plot.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that all the points are tightly hugged by a trend line
\nHence, from the above,
\nWe can conclude that the given scatter plot has a stronger association<\/p>\n
Question 5.
\nDescribe the association between the two sets of data in the scatter plot.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that the points are all in a non-linear shape and are closely connected
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association<\/p>\n
Practice & Problem Solving<\/strong><\/p>\nScan for Multimedia<\/p>\n
Question 6.
\nThe scatter plot shows the average heights of children ages 2-12 in a certain country. Which line is the best model of the data?
\n
\nAnswer:
\nIt is given that
\nThe scatter plot shows the average heights of children ages 2-12 in a certain country.
\nNow,
\nThe best line in the given scatter plot is that line that tightly hugs the maximum points in a scatter plot
\nHence, from the above,
\nWe can conclude that line m is the best model of the given data<\/p>\n
Question 7.
\nDoes the scatter plot shows a positive, a negative, or no association?
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that as the value of x increases, the value of y also increases
\nHence, from the above,
\nWe can conclude that the given scatter plot has a positive association<\/p>\n
Question 8.
\nDetermine whether the scatter plot of the data for the following situation would have a positive or negative linear association.
\ntime working and amount of money earned
\nAnswer:
\nThe given situation is:
\nTime working and amount of money earned
\nWe know that,
\nThe total amount of work done = Number of days \u00d7 The amount earned for the work done
\nLet us suppose the number of days is constant
\nSo,
\nThe total amount of work done \u221d The amount earned for the work done
\nSo,
\nThe more time a person works, the more money that person will earn
\nHence, from the above,
\nWe can conclude that the scatter plot of the given data has a positive linear association<\/p>\n
Question 9.
\nDescribe the relationship between the data in the scatter plot.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that the data in the scatter plot has a decreasing trend with the strong association of data with each other
\nHence, from the above
\nWe can conclude that the given scatter plot has a negative linear association<\/p>\n
Question 10.
\nDescribe the relationship between the data in the scatter plot.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that all the points are in a cyclic fashion
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association<\/p>\n
Question 11.
\nHigher-Order Thinking<\/strong> Describe a real situation that would fit the relationship described.
\na. A strong, positive association
\nAnswer:
\nThe real-life examples for a strong, positive association are:
\nA) The more time you spend running on a treadmill, the more calories you will burn.
\nB) Taller people have larger shoe sizes and shorter people have smaller shoe sizes.
\nC) The longer your hair grows, the more shampoo you will need.
\nD) The less time I spend marketing my business, the fewer new customers I will have.
\nE) The more hours you spend in direct sunlight, the more severe your sunburn.<\/p>\nb. A strong, negative association
\nAnswer:
\nThe real-life examples for a strong, negative association are:
\nA) A student who has many absences has a decrease in grades.
\nB) As the weather gets colder, air conditioning costs decrease.
\nC) If a train increases speed, the length of time to get to the final point decreases.
\nD) If a chicken increases in age, the number of eggs it produces decreases.
\nE) If the sun shines more, a house with solar panels requires less use of other electricity.<\/p>\n
Question 12.
\nA sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study. What type of association does it show between the amount of sleep and money spent?
\n
\nAnswer:
\nIt is given that
\nA sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the data that is related to the amount of sleep and the amount of money spent is in a cyclic fashion
\nHence, from the above,
\nWe can conclude that the association does it show between the amount of sleep and money spent is: Non-linear association<\/p>\n
Assessment Practice<\/strong><\/p>\nQuestion 13.
\nWhich paired data would likely show a positive association? Select all that apply.
\n Population and the number of schools
\n Hair length and shoe size
\n Number of people who carpool to work and money spent on gas
\n Hours worked and amount of money earned
\n Time spent driving and amount of gas in the car
\nAnswer:
\nWe know that,
\nA positive association is an association that as the value of x increases, the value of y also increases
\nHence,
\nThe paired data that would likely show a positive association is<\/p>\n
Question 14.
\nWhich paired data would likely show a negative association? Select all that apply.
\n Population and the number of schools
\n Hair length and shoe size
\n Number of people who carpool to work and money spent on gas
\n Hours worked and amount of money earned
\n Time spent driving and amount of gas in the car
\nAnswer:
\nWe know that,
\nA negative association is an association that as the value of x increases, the value of y also decreases
\nHence,
\nThe paired data that would likely show a negative association is:
\n<\/p>\n
Lesson 4.3 Use Linear Models to Make Predictions<\/h3>\n
Solve & Discuss It!<\/strong><\/p>\nBao has a new tracking device that he wears when he exercises. It sends data to his computer. How can Bao determine how long he should exercise each day if he wants to burn 5,000 Calories per week?
\n
\nAnswer:
\nIt is given that
\nBao has a new tracking device that he wears when he exercises. It sends data to his computer
\nNow,
\nIt is also given that Bao wants to burn 5,000 calories per week
\nSo,
\nThe number of calories Bao wants to burn per day = \\(\\frac{5,000}{7}\\)
\n= 714.2 calories
\n= 714 calories
\n\u2245 720 calories
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nFor approximately 720 calories to burn, Bao has to exercise 80 – 90 minutes each day
\nHence, from the above,
\nWe can conclude that Bao should exercise 80 – 90 minutes each day if he wants to burn 5,000 Calories per week<\/p>\n
Focus on math practices<\/strong><\/p>\nReasoning<\/strong> Suppose another set of data were plotted with a trend line passing through (25, 100) and (80, 550). Would this indicate that more or fewer calories were burned per minute? Explain.
\nAnswer:<\/p>\n? Essential Question<\/strong>
\nHow do linear models help you to make a prediction?
\nAnswer:
\nWhile\u00a0linear models\u00a0do not always accurately represent\u00a0data, and this occurs when actual\u00a0data\u00a0does not clearly show a relationship between its two variables,\u00a0linear models\u00a0are\u00a0helpful\u00a0in determining the future points of\u00a0data, the expected points of\u00a0data, and the highest possible accuracy of\u00a0data.<\/p>\nTry It!<\/strong><\/p>\nAssuming the trend shown in the graph continues, use the equation of the trend line to predict average fuel consumption in miles per gallon in 2025.
\nThe equation of the trend line is y = x + . In 2025, the average fuel consumption is predicted to be about mpg.
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the scatter plot,
\nWe can observe that
\nThe initial value (y-intercept) is: 15
\nNow,
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nTo find the slope,
\nThe points from the given scatter plot is: (15, 21), (30, 24)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{24 – 21}{30 – 15}\\)
\n= \\(\\frac{1}{5}\\)
\nSo,
\nThe equation of the line is:
\ny = 0.20x + 15
\nNow,
\nThe average fuel consumption in 2025 is:
\ny = 0.20 (2025 – 1980) + 15
\ny = 0.20 (45) + 15
\ny = 9 + 15
\ny = 24 mpg
\nHence, from the above,
\nWe can conclude that
\nThe equation of the trend line is:
\ny = 0.20x + 15
\nIn 2025, the average fuel consumption is predicted to be about 24 mpg<\/p>\nConvince Me!<\/strong>
\nWhy can you use a linear model to predict the y-value for a given x-value?
\nAnswer:
\nWe can use the regression line to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the\u00a0value of Y. The\u00a0predicted value of Y\u00a0is called the\u00a0predicted value of Y, and is denoted\u00a0Y’.<\/p>\nTry It!<\/strong><\/p>\nA smoothie caf\u00e9 has the ingredients needed to make 50,000 smoothies on a day when the high temperature is expected to reach 90\u00b0F. Should the caf\u00e9 employees expect to have enough ingredients for the day’s smoothie sales? Explain.
\nAnswer:<\/p>\n
KEY CONCEPT<\/strong>
\nScatter plots can be used to make predictions about current or future trends.
\nLook for the corresponding y-value for a given x-value.
\n<\/p>\nFind the equation of the trend line and find the y-value of a given x-value.
\n<\/p>\n
Do You Understand?<\/strong><\/p>\nQuestion 1.
\n?Essential Question<\/strong> How do linear models help you to make a prediction?
\nAnswer:
\nWhile\u00a0linear models\u00a0do not always accurately represent\u00a0data, and this occurs when actual\u00a0data\u00a0does not clearly show a relationship between its two variables,\u00a0linear models\u00a0are\u00a0helpful\u00a0in determining the future points of\u00a0data, the expected points of\u00a0data, and the highest possible accuracy of\u00a0data.<\/p>\nQuestion 2.
\nModel with Math<\/strong>
\nHow do you find the equation of a linear model when you are given the graph but not given the equation?
\nAnswer:
\nTo simplify what has already been said, the easiest way to find the equation of a line is to look for the x and y-intercepts.
\nOne point will be (x, 0) and the other will be (0, y), where x and y are numerical values.
\nThe slope is simply
\nm = \\(\\frac{y}{x}\\).
\nWhen you have the y-intercept, (0, y),
\nyou can use the form
\ny = mx + b to find the equation for the line.
\nConsequently, with the notation used, you can represent this as
\ny=\\(\\frac{y}{x}\\)x + b
\nwhere b is the value from (0, y)
\nx is the value from (x, 0)<\/p>\nQuestion 3.
\nReasoning<\/strong> Can the linear model for a set of data that is presented in a scatter plot always be used to make a prediction about any x-value? Explain.
\nAnswer:
\nYes, we can use the linear model to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the value of Y. The predicted value of Y is called the predicted value of Y, and is denoted Y’.<\/p>\nDo You Know How?<\/strong><\/p>\nQuestion 4.
\nThe graph shows a family’s grocery expenses based on the number of children in the family,
\n
\na. Using the slope, predict the difference in the amount spent on groceries between a family with five children and a family with two children.
\nAnswer:
\nIt is given that
\nThe graph shows a family’s grocery expenses based on the number of children in the family,
\nNow,
\nThe given scatter plot is:
\n
\nNow,
\nFrom the given scatter plot,
\nThe pair that represents the amount spent on groceries in a family with five children is: (5, 175)
\nThe pair that represents the amount spent on groceries in a family with two children is: (2, 140)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{140 – 175}{2 – 5}\\)
\nSlope = \\(\\frac{-35}{-3}\\)
\nSlope = \\(\\frac{35}{3}\\)
\nHence, from the above,
\nWe can conclude that using the slope, the difference in the amount spent on groceries between a family with five children and a family with two children is: \\(\\frac{35}{3}\\)<\/p>\nb. How many children can you predict a family has if the amount spent on groceries per week is $169.47?
\nAnswer:
\nFrom the given scatter plot,
\nThe trend line equation is:
\ny = 21.08x + 85.15
\nNow,
\nIt is given that the amount spent on groceries per week is $169.47
\nSo,
\n169.47 = 21.08x + 85.15
\n21.08x = 169.47 – 85.15
\n21.08x = 84.32
\nx = \\(\\frac{84.32}{21.08}\\)
\nx = 4
\nHence, from the above,
\nWe can predict 4 children in a family if he has the amount spent on groceries per week is $169.47<\/p>\n
Practice & Problem Solving<\/strong><\/p>\nQuestion 5.
\nLeveled Practice<\/strong> The scatter plot shows the number of people at a fair based on the outside temperature. How many fewer people would be predicted to be at the fair on a 100\u00b0F day than on a 75\u00b0F day?
\nThe slope is
\nFor each degree that the outside temperature increases, the fair attendance decreases by thousand people.
\n
\nThe difference between 75\u00b0F and 100\u00b0F is \u00b0F.
\n-0.16 . =
\nAbout thousand fewer people are predicted to be at the fair on a 100\u00b0F day than on a 75\u00b0F day.
\nAnswer:
\nIt is given that
\nThe scatter plot shows the number of people at a fair based on the outside temperature
\nNow,
\nThe given scatter plot is:
\n
\nNow,
\nWe know that,
\nThe equation of the trend line that is passing through two points is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nTo find the slope,
\nThe given points are: (75, 10K), (100, 6K)
\nWhere,
\nK is 1000
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{6K – 10K}{100 – 75}\\)
\n= –\\(\\frac{4,000}{25}\\)
\n= -160
\nSo,
\nThe equation of the trend line is:
\ny = -160x + b
\nSubstitute (100, 6K) in the above equation
\nSo,
\n6K = -160 (100) + b
\n6K + 16K = b
\nb = 22K
\nSo,
\nThe equation of the trend line is:
\ny = -160x + 22,000
\nNow,
\nAt 75\u00b0 F,
\ny = -160 (75) + 22,000
\ny = 10,000
\nAt 100\u00b0 F,
\ny = -160 (100) + 22,000
\ny = 6,000
\nHence,
\nThe difference of the people between 75\u00b0 F and 100\u00b0 F = 10,000 – 6,000
\n= 4,000
\nThe difference between 100\u00b0 F and 75\u00b0 F = 25\u00b0 F<\/p>\nQuestion 6.
\nMake Sense and Persevere<\/strong> If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas prices in 2013 and 2001 is? Round to the nearest hundredth.
\n
\nAnswer:
\nIt is given that
\nx represents the number of years since 2000 and y represents the gas price,
\nNow,
\nWe know that,
\nThe equation of the trend line in the slope-intercept form is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept
\nNow,
\nTo find the slope of the trend line,
\nThe given points are: (7, 3), (12, 4)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{4 – 3}{12 – 7}\\)
\nSlope = \\(\\frac{1}{5}\\)
\nSo,
\ny = \\(\\frac{1}{5}\\)x + b
\nNow,
\nSubstitute (7, 3) or (12, 4) in the above equation
\nSo,
\n3 = \\(\\frac{1}{5}\\) (7) + b
\n\\(\\frac{8}{5}\\) = b
\nSo,
\nThe equation of the trend line is:
\n5y = x + 8
\nNow,
\nThe gas prices in 2001 is:
\n5y = 1 + 8
\ny = \\(\\frac{9}{5}\\)
\ny = $1.40
\nThe gas prices in 2013 is:
\n5y = 13 + 8
\ny = \\(\\frac{21}{5}\\)
\ny = $4.20
\nSo,
\nThe difference between the gas prices in 2013 and 2001 = $4.20 – $1.40
\n= $2.80
\nHence, from the above,
\nWe can conclude that the difference between the gas prices in 2013 and 2001 is: $2.80<\/p>\nQuestion 7.
\nMake Sense and Persevere<\/strong> If x represents the number of months since the beginning of 2016, and y represents the total precipitation to date, predict the amount of precipitation received between the end of March and the end of June.
\n
\nAnswer:
\nIt is given that
\nx represents the number of months since the beginning of 2016, and y represents the total precipitation to date
\nNow,
\nThe given scatter plot is:
\n
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the trend line starts from the origin
\nSo,
\nThe equation of the trend line that is passing through the origin is:
\ny = mx
\nwhere,
\nm is the slope
\nNow,
\nTo find the slope.
\nThe given points are: (2, 10), (10, 40)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{40 – 10}{10 – 2}\\)
\n= \\(\\frac{30}{8}\\)
\n= \\(\\frac{15}{4}\\)
\nSo,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{4}\\)x
\nNow,
\nAt the end of the march,
\nThe amount of precipitation is:
\ny = \\(\\frac{15}{4}\\) (4)
\ny = 15 in
\nAt the end of June,
\nThe amount of precipitation is:
\ny = \\(\\frac{15}{4}\\) (6)
\ny = \\(\\frac{45}{2}\\)
\ny = 22.5 inches
\nSo,
\nThe amount of precipitation between the end of March and the end of June = 22.5 – 15
\n= 7.5 in
\nHence, from the above,
\nWe can conclude that the amount of precipitation between the end of March and the end of June is: 7.5 in<\/p>\nQuestion 8.
\nThe scatter plot shows a hiker’s elevation above sea level over time. The equation of the trend line shown is y = 8.77x + 686. To the nearest whole number, predict what the hiker’s elevation will be after 145 minutes.
\n
\nAnswer:
\nIt is given that
\nThe scatter plot shows a hiker’s elevation above sea level over time.
\nThe equation of the trend line shown is
\ny = 8.77x + 686.
\nWhere,
\n8.77 is the slope
\n686 is the initial value (or) y-intercept
\nNow,
\nFrom the graph,
\nWe can observe that
\nThe x-axis variable – Time
\nThe y-axis variable – Elevation
\nSo,
\nThe hiker’s elevation after 145 minutes is:
\ny = 8.77 (145) + 686
\ny = 1,957.65 ft
\nHence, from the above,
\nWe can conclude that the hiker’s elevation after 145 minutes will be: 1,957.65 ft<\/p>\n
Question 9.
\nMake Sense and Persevere<\/strong> The graph shows the number of gallons of water in a large tank as it is being filled. Based on the trend line, predict how long it will take to fill the tank with 375 gallons of water.
\n
\nAnswer:
\nIt is given that
\nThe graph shows the number of gallons of water in a large tank as it is being filled
\nNow,
\nThe given scatter plot is:
\n
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nThe initial value (or) y-intercept is: 15
\nNow,
\nWe know that,
\nThe equation of the trend line that has the initial value is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept (or) initial value
\nNow,
\nTo find the slope,
\nThe required points are: (1, 30), (0, 15)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{15 – 30}{0 – 1}\\)
\n= \\(\\frac{-15}{-1}\\)
\n= 15
\nSo,
\nThe equation of the trend line is:
\ny = 15x + 15
\nNow,
\nThe time taken to fill 375 gallons of water is:
\n375 = 15x + 15
\n15x = 375 – 15
\n15x = 360
\nx = \\(\\frac{360}{15}\\)
\nx = 24 minutes
\nHence, from the above,
\nWe can conclude that the time taken to fill 375 gallons of water is: 24 minutes<\/p>\nQuestion 10.
\nHigher-Order Thinking<\/strong> The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes?
\n
\nAnswer:
\nIt is given that
\nThe graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows.
\nNow,
\nFrom the given scatter plot,
\nWe can observe that,
\nAt 5 minutes of time, the freezer temperature is 15\u00b0F
\nSo,
\nAt x = 5, y = 15
\nBut,
\nAccording to the given information
\nAt x = 5, y = 15 + 3
\nSo,
\ny = 18\u00b0F
\nHence, from the above,
\nWe can conclude that the actual temperature after the freezer was on for five minutes is: 18\u00b0F<\/p>\nAssessment Practice<\/strong><\/p>\nQuestion 11.
\nThe graph shows the altitude above sea level of a weather balloon over time. The trend line passes through the points (0, 453) and (10, 359). Which statements about the graph are true?
\n The data show a positive correlation.
\n The trend line is -9.4x – 453.
\n In general, the balloon is losing altitude.
\n The weather balloon started its flight at about 455 feet above sea level.
\n
\n After 4 minutes, the weather balloon had an altitude of about 415 feet above sea level.
\n After 395 minutes, the weather balloon had an altitude of about 8 feet above sea level.
\nAnswer:
\nLet the given options be named as A, B, C, D, E and F respectively
\nIt is given that
\nThe graph shows the altitude above sea level of a weather balloon over time.
\nThe trend line passes through the points (0, 453) and (10, 359)
\nWe know that,
\nThe equation of the trend line that is passing through two points is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nWe know that
\nThe “y-intercept” is the value of y when x= 0
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{359 – 453}{10 – 0}\\)
\n= \\(\\frac{-94}{10}\\)
\n= -9.4
\nSo,
\nThe equation of the trend line is:
\ny = -9.4x + 453
\nHence, from the above,
\nWe can conclude that options C, D, and E matches the given situation<\/p>\nTopic 4 MID-TOPIC CHECKPOINT<\/h3>\n
Question 1.
\nVocabulary<\/strong> How can you use a trend line to determine the type of linear association for a scatter plot? Lesson 4.2
\nAnswer:
\nThe straight\u00a0line\u00a0is a\u00a0trend line, designed to come as close as possible to all the data points. The\u00a0trend line\u00a0has a positive slope, which shows a positive\u00a0relationship\u00a0between X and Y. The points in the\u00a0graph\u00a0are tightly clustered about the\u00a0trend line\u00a0due to the strength of the\u00a0relationship\u00a0between X and Y.<\/p>\nThe scatter plot shows the amount of time Adam spent studying and his test scores. Use the scatter plot for Items 2-4.<\/p>\n
Question 2.
\nWhat relationship do you see between the amount of time spent studying and the test scores? Is the relationship linear? Lesson 4.1
\nA. In general, Adam scores higher on a test when he spends more time studying. There is not a linear relationship.
\nB. In general, Adam scores higher on a test when he spends more time studying. There is a positive linear relationship.
\n
\nC. In general, Adam scores lower on a test when he spends more time studying. There is a negative linear relationship.
\nD. In general, Adam scores lower on a test when he spends more time studying. There is no relationship.
\nAnswer:
\nIt is given that
\nThe scatter plot shows the amount of time Adam spent studying and his test scores
\nNow,
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that
\nThe association or correlation is positive and there is a linear relationship
\nAdam is scoring higher on a test when he is studying for more hours
\nHence, from the above,
\nWe can conclude that option B matches the given situation<\/p>\n
Question 3.
\nUse the y-intercept and the point (4,90) from the line on the scatter plot. What is the equation of the linear model? Lesson 4.3
\nAnswer:
\nThe given scatter plot is (From Question 2):
\n
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nThe initial value (or) y-intercept is: 60
\nWe know that,
\nThe “y-intercept” is the value of y when x = 0
\nSo,
\nThe points required to find the equation of the scatter plot is: (0, 60), (4, 90)
\nWe know that,
\nThe equation of the trend line that has y-intercept is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{90 – 60}{4 – 0}\\)
\n= \\(\\frac{30}{4}\\)
\n= \\(\\frac{15}{2}\\)
\nSo,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{2}\\)x + 60
\nHence, from the above,
\nWe can conclude that the equation of the linear line is:
\ny = \\(\\frac{15}{2}\\)x + 60<\/p>\nQuestion 4.
\nPredict Adam’s test score when he studies for 6 hours. Lesson 4.3
\nAnswer:
\nWe know that,
\nFrom the given scatter plot,
\nThe variable on the x-axis is: Time
\nThe variable on the y-axis is: Test scores
\nNow,
\nFrom Problem 3,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{2}\\)x + 60
\nAt 6 hours,
\ny = \\(\\frac{15}{2}\\) (6) + 60
\ny = 105
\nHence, from the above,
\nWe can conclude that Adam’s test score is 105 when he studies for 6 hours<\/p>\n
Question 5.
\nDescribe the relationship between the data in the scatter plot. Lesson 4.2
\n
\nAnswer:
\nThe given scatter plot is:
\n
\nFrom the above scatter plot,
\nWe can observe that as the value of x increases, the value of y decreases
\nHence, from the above,
\nWe can conclude that the given scatter plot has the negative association<\/p>\n
Question 6.
\nThe scatter plot shows the mean annual temperature at different elevations. Select all the observations that are true about the scatter plot. Lesson 4.1
\n The majority of the elevations are in a cluster between 1,250 meters and 2,250 meters.
\n There is a gap in the data between 500 meters and 1,250 meters.
\n
\n There is an outlier at about (50, 21).
\n In general, the mean annual temperature decreases as the elevation increases.
\n Because there is a gap in the values, there is no association between the temperature and elevation.
\nAnswer:
\nLet the given options be named as A, B, C, D, and E
\nIt is given that
\nThe scatter plot shows the mean annual temperature at different elevations
\nNow,
\nThe given scatter plot is:
\n
\nFrom the given scatter plot,
\nWe can observe that
\nThere is a cluster between 1,250 m and 2,250m
\nThere is a gap between 500m and 1,250m
\nIn general, the mean annual temperature decreases as the elevation increases.
\nBecause there is a gap in the values, there is no association between the temperature and elevation.
\nHence, from the above,
\nWe can conclude that A, B, D, and E matches the given situation
\nHow well did you do on the mid-topic checkpoint? Fill in the stars. <\/p>\n
Topic 4 MID-TOPIC PERFORMANCE TASK<\/h3>\n
A pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.
\n<\/p>\n
PART A<\/strong>
\nConstruct a scatter plot of the data in the table.
\nAnswer:
\nIt is given that
\nA pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.
\nNow,
\nThe given table is:
\n
\nHence,
\nThe representation of the scatter plot for the given data is:
\n<\/p>\n