Lesson 6.2.1

HW: Day 1: Problems 6-55 through 6-60

Day 2: Problems 6-61 through 6-66

Learning Target: Scholars will observe the impact of an outlier on the LSRL. Scholars will determine if a linear model is a good fit for the data by creating and visually analyzing residual plots.

In this lesson, you will create residual plots to help you visually determine how well a LSRL fits the data.

6-47. In problem 6-1, you completed an investigation that helped Robbie use a viewing tube to see a football game. Typical data is shown in the table below.

  1. Use your calculator or6-47 Student eTool(Desmos)to create a scatterplot and LSRL. Sketch the graph on your paper. What is the equation of the LSRL?
  2. When entering the data in her calculator, Amy accidentally entered (144,10.7) for the first data point. Make this change to your data and sketch the new point and new LSRL in a different color. Would you consider this point an outlier?
  3. What is the impact of the outlier? Will Amy’s predictions for the field of view be too large or too small? How do you know?

6-48. Giulia’s father would like to open a restaurant, and is deciding how much to charge for the toppings on pizza. He sent Giulia to eight different Italian restaurants around town to find out how much they each charge. Giulia returned with the following information:

  1. Giulia needs to write a report for her father about what he should charge for a two-topping pizza. Discuss with your team what elements a statistical analysis report should contain. Explore your ideas using the6-48 & 6-50 Student eTool(Desmos) .
  2. Write the report you described in part (a), and predict what Giulia’s father should charge for a two-topping pizza.

6-49.A residual plot can help you determine if a linear model is a good fit for a scatterplot of data. Whenever a LSRL line is drawn on a scatterplot, a residual plot can also be created. A residual plot has an x-axis that is the same as the scatterplot, and a y-axis that plots the residual.

  1. Match each scatterplot below with its corresponding residual plot.
  2. For which of the scatterplots does a linear model fit the data best? How does the residual plot help you make that decision?

6-50.Help Giulia analyze the residuals for the pizza parlors in problem 6-48 as described below. Explore ideas using the6-48 & 6-50 Student eTool(Desmos) .

  1. Mark the residuals on the scatterplot you created in problem 6-48. If you want to purchase an inexpensive pizza, should you go to a store with a positive or negative residual?
  2. What is the sum of the residuals? Are you surprised at this result?
  3. Your teacher will show you how to make a residual plot using your calculator, with the x-axis representing the number of pizza toppings, and the y-axis representing the residuals. Interpret the scatter of the points on the residual plot. Is a linear model a good fit for the data?

6-51. Dry ice (frozen carbon dioxide) evaporates at room temperature. Giulia’s father uses dry ice to keep the glasses in the restaurant cold. Since dry ice evaporates in the restaurant cooler, Giulia was curious how long a piece of dry ice would last. She collected the following data:

  1. Sketch the scatterplot and LSRL of this data.
  2. Use your calculator or6-51 Student eTool(Desmos)to make a residual plot to determine if a linear model is appropriate. Make a conjecture about what the residual plot tells you about the shape of the original data Giulia collected.

6-55.In problem 6-24 you looked at the data for a study conducted on a vitamin supplement that claims to shorten the length of the common cold. The data is repeated in the table below:

  1. Find the LSRL. Create a scatterplot on graph paper (or use your scatterplot from problem6-24) and draw the line of best fit. What is the equation of the line of best fit?
  2. Draw the upper and lower boundary lines on the graph following the process you used in problem 6-23. What is the equation of the upper boundary line? Of the lower boundary line?
  3. Based on the upper and lower boundary lines of your model, what do you predict is the length of a cold for a person who has taken the supplement for 3months? Consider the precision of the data and use an appropriate number of decimal places in your response.
  4. How long do you predict a cold will last for a person who has taken no supplement? Interpret the y-intercept in context.
  5. How long do you predict a cold will last for a person who has taken 6 months of supplements?
  6. If you have a cold, would you prefer a negative or positive residual?

6-56.Write an explicit equation for the sequence graphed below.

o 

6-57. Graph the points (−3, 2) and (5, −4). 6-57 HW eTool (Desmos).

  1. Find the slope of the line determined by them.
  2. Find the distance between them.
    (Hint: Pythagorean theorem)

6-58. Graph y = (x − 2)2 − 1. Completely describe the graph. 6-58 HW eTool (Desmos).

6-59.Copy each generic rectangle below and fill in the missing parts. Then write an equation showing the area as a product and the area as a sum.

6-60. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems is shown at right.
a.
/ b.
/ c.
/ d.

6-61. Solve each equation below. Check each solution.
  1. 6 − (3 + x) = 10
  2. 100(x + 3) = 200

6-62. Write an equation or system of equations to solve.

·  Thanh bought 11 pieces of fruit and spent $5.60. If apples cost $0.60 each and pears cost $0.35 each, how many of each kind of fruit did he buy?

6-63.Find the slope of the line through the points (6, −8) and (3, −4). 6-63 HW eTool (Desmos).

  1. Find the equation of the line through (6, −8) and (3, −4).
  2. Is the point (−3, 4) on the line you found in part (a)? How can you tell?

6-64.Dorinae is confused. She is making a x→ y table. She is trying to find y when x = –3,but she is not sure if she should find the absolute value first, or if she should first add 1. Explain to Dorinae what she should do first. Justify your reasoning.

6-65.Solve this system of equations:

o 

  1. What does your solution tell you about the relationship between the lines?
  2. Solve the second equation for y.
  3. Does the slope of each line confirm your statement in part (a)? Explain how.

6-66.Figure 3 of a tile pattern is shown at right. If the pattern grows linearly and if Figure 7 has 13 tiles, then find an equation for the pattern.

Lesson 6.2.1

·  6-47. See below:

  1. y = 1.66 + 0.13x
  2. y = 7.49 + 0.06x; Yes; the new point is far away from the pattern of the rest of the data.
  3. The LSRL is “pulled” toward the outlier. Amy’s mistake will cause predictions close to the wall to be too large, and predictions far from the wall to be too small.

·  6-48. See below:

  1. A scatterplot, a LSRL on the scatterplot, an equation for the LSRL, a description of the association (form, direction strength, and outliers), and possibly a description of the bounds.
  2. See scatterplot below.y = 7.74 + 1.36x; form: linear, direction: positive with a slope of 1.36 meaning an increase of one topping is expected to increase the cost by $1.36, strength: strong. About $10.46 for a two-topping pizza.

·  6-49. See below:

  1. I-C, II-B, III-A
  2. Scatterplot II. If the data in the residual plot appears to be randomly scattered, with no pattern, the linear model fits the data well. The curved Residual Plot A for Scatterplot III indicates a curved line of best fit would be better. The fan-shaped Residual Plot C for Scatterplot I indicates that as the x-values get larger, there is more and more variability in the observed data; predictions made from smaller x-values will probably be closer to the observed value than predictions made from larger xvalues.

·  6-50. See below:

  1. Negative residual is lower cost than predicted.
  2. 0; No, the LSRL goes through the “middle” of all the data points, so we expect the positive residuals to equal the negative residuals.
  3. There does not appear to be any kind of shape or pattern to the plotted points. That means the model fits through the data points well. That is, our LSRL linear model is appropriate.

·  6-51. See below:

  1. See graph below. y=15.21–0.52x
  2. Residual plot shows a clear curve; original data was curved, probably exponentially.

·  6-52. See below:

  1. Yes. The scatter appears random and there is no apparent pattern in the residual plot.
  2. 9.37+3.96(62.9)=258
  3. The residual is about –18, so the actual observed is about 240.

·  6-53. Points are not evenly scattered around the plot. Points can be in clusters and there can be large gaps, however, there is no discernable trend or pattern to the points.

·  6-55. See below:

  1. y = 5.37 − 1.58x
  2. y = 6.16 − 1.58x and y = 4.58 −1.58x, based ona maximum residual of −0.79.
  3. 0 to 1.4 days. The measurements had one decimal place.
  4. Between 4.6 and 6.2 days. The y-intercept is the number of days a cold will last for a person who takes no supplements.
  5. Students should predict that a negative number of days makes no sense here. Statistical models often cannot be extrapolated far beyond the edges of the data.
  6. A negative residual is desirable because it means the actual cold was shorter than was predicted by the model.

·  6-56. an = t(n) = 32()n

·  6-57. See below:

  1. −
  2. 10

·  6-58. See graph below. The graph is a parabola opening upward. From left to right the graph decreases until x = 2 and then increases. The vertex is at (2, −1). The xintercepts are at (1, 0) and (3, 0). The y-intercept is at (0, 3). The line of symmetry is at x = 2. The domain is all real numbers and the range is y≥−1 .

·  6-59. See below:

  1. (5x − 3)(2x − 4y + 5) = 10x2 − 20xy + 19x + 12y − 15
  2. Likely answers include (x + 12)(x + 1) = x2 + 13x + 12, (x + 6)(x + 2) = x2 + 8x + 12, and (x + 4)(x + 3) = x2 + 7x + 12, although other answers are possible.

·  6-60. See answers in bold in the diamonds below:

·  6-61. See below:

  1. x = −7
  2. x = −1
  3. x = 9
  4. x = 34

·  6-62. a + p = 11, 0.60a + 0.35p = $5.60; 7 apples and 4 pears

·  6-63. −

  1. y = −x
  2. Yes; Substitute –3 for x and 4 for y.

·  6-64. She should add 1 first, since the addition is placed inside the absolute value symbol, which acts as a grouping symbol.

·  6-65. See below:

  1. There is no solution, so the lines do not intersect.
  2. y = x −
  3. Yes; both lines have the same slope.

·  6-66. y = 2x − 1