Bivariate Data

Lesson 6

Objective

Interpret the slope and $$y$$-intercept of a fitted line in context.

Common Core Standards

Core Standards

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  • 8.SP.A.3 — Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

Foundational Standards

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  • 8.F.B.4

Criteria for Success

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  1. Distinguish between linear functions where an equation exactly models a relationship and a linear model for a scatter plot where an equation represents an association between two variables.
  2. Given an equation for a line that represents an association between bivariate data, interpret the slope and $$y$$-intercept in context of the data (MP.2).
  3. Use an equation to solve problems with bivariate data.

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Anchor Problems

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Problem 1

At a restaurant, the amount of tip for the waitress or waiter is automatically calculated at 20% of the bill total. The graph below shows the amount a tip would be for 6 different bill totals. 

  1. Write an equation to represent the amount of tip based on the bill total. 
  2. What is the rate of change and what does it represent in the context of the problem?
  3. What is the initial value and what does it represent in the context of the problem?
  4. If you had a bill that came to $75.80, then how much money will you pay for the tip?

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Problem 2

A different restaurant does not automatically calculate the tip amount, but rather, the customers determine how much they want to leave as a tip. The graph below shows the amount of tip that 14 different customers left, based on the amount of their bill. 

A line has been drawn to represent the trend in data. The equation for the line is: $${y=0.14x+2.5}$$.

  1. What does the 0.14 mean in this situation?
    1. The average tip amount is $0.14.
    2. On a $1 bill, the tip amount is estimated at $0.14.
    3. A $1 increase in the bill total is associated with a $0.14 increase in tip amount.
    4. A $0.14 increase in the bill total is associated with a $1 increase in tip amount.

 

  1. What does the 2.5 mean in this situation?
    1. The average bill amount is $2.50.
    2. The equation predicts a $2.50 tip on a $0 bill.
    3. The equation predicts a $0.14 tip on a $2.50 bill.
    4. A $1 increase in the bill total is associated with a $2.50 increase in tip amount.

 

  1. If a bill is $75.80, what prediction can a waiter or waitress make about the amount of the tip?

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Problem 3

After giving a test, a teacher was curious to know if there was a relationship between how a student performed on the test and how long the student spent studying. She collected data from her classes and represented it in the scatter plot below.

She found a line that best fit the data and determined the equation for the line to be: $${y=0.8x+59}$$.  Which of the statements below are fair conclusions that she can make about her model? Select all that apply.

  1. A student who did not study for the test is predicted to earn a grade of 59 points.
  2. A student who studied for 59 minutes is predicted to earn a grade of 80 points.
  3. Each additional minute of study time is associated with an additional 0.8 points on the test.
  4. Each additional minute of study time is associated with an additional 59 points on the test.
  5. There is a positive linear relationship between the two variables because as the study time increases, then the test grade increases.

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Target Task

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According to the Bureau of Vital Statistics for the New York City Department of Health and Mental Hygiene, the life expectancy at birth (in years) for New York City babies is shown in the scatter plot below.

An equation for a line fit to this data is represented by: $${y=0.338x-597.4}$$

  1. Explain what the slope of this equation model means in terms of the context. 
  2. Use the model to predict the life expectancy for a baby born in New York City in 2020.

References

EngageNY Mathematics Grade 8 Mathematics > Module 6 > Topic C > Lesson 11Exit Ticket

Grade 8 Mathematics > Module 6 > Topic C > Lesson 11 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

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