Unit 6, Lesson 6
The Slope of a Fitted Line
Grade 8
6.1
Warm-up
Estimate the slope of the line. Be prepared to explain your reasoning.
6.2
Activity
For each scatter plot, decide if there is an association between the 2 variables, and describe the situation using one of these sentences:
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For these data, as ________________ increases, ________________ tends to increase.
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For these data, as ________________ increases, ________________ tends to decrease.
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For these data, ________________ and ________________ do not appear to be related.
A scatterplot. Horizontal, from 0 to 150000, by 30000's labeled mileage. Vertical, from 6000 to 21,000, by 3000's abeled price. 18 data points trend linearly downward and right.
A scatterplot. Horizontal, from 10 to 50, by fives, labeled initial battery life (hours). Vertical, from 10 to 45 by fives, labeled later battery life (hours). 20 data points trend linearly upward and right.
A scatterplot. Horizontal, from 70 to 105, by 5's, labeled high temperature, Farenheit. Vertical, from 15 to 45, by 5's, labeled energy consumed, kilowatt hours. 28 dots, trend linearly upward and right.
6.3
Activity
For each of the situations, a linear model for some data is shown.
- What is the slope of the line in the scatter plot for each situation?
- What is the meaning of the slope in that situation?
A scatterplot. Horizontal, from 10 to 50, by fives, labeled initial battery life (hours). Vertical, from 10 to 45, by fives, labeled later battery life (hours). Line drawn, trend linearly upward and right with 20 data points above and below line.
A scatterplot. Horizontal, from 1000 to 2500, by 250’s, labeled weight, kilograms. Vertical, from 14 to 32, by 2’s, labeled fuel efficiency, miles per gallon. 20 dots, trend linearly downward and right, dot at 1125 comma 28 and dot at 1950 comma 19 with line of best fit.
A scatterplot. Horizontal, from 70 to 105, by 5’s, labeled high temperature, Farenheit. Vertical, from 15 to 45, by 5’s, labeled energy consumed, kilowatt hours. 28 dots, trend linearly upward and right, dot at 86 comma 29 and dot at 98 comma 36 with line of best fit.
6.4
Activity
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For each of the scatter plots, decide whether it makes sense to fit a linear model to the data. If it does, would the graph of the model have a positive slope, a negative slope, or a slope of 0?
A A scatterplot, points begin near 0 comma 40 and trend down and right.B A scatterplot, points begin around the origin and trend up and right.C A scatterplot, begins near 0 comma 500 and trend down and right.D A scatterplot, points are distributed all around the graph.E A scatterplot, points begin around 22 comma 9 and trend up and right, with a single point near 24 comma 7. - Which of these scatter plots show evidence of a positive association between the variables? Of a negative association? Which do not appear to show an association?
Student Lesson Summary
Here is a scatter plot that we have seen before. As noted earlier, we can see from the scatter plot that taller dogs tend to weigh more than shorter dogs.
Another way to say it is that weight tends to increase as height increases.
When we have a positive association between two variables, an increase in one means there tends to be an increase in the other.
A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points. Trend upward and to right.
We can quantify this tendency by fitting a line to the data and finding its slope.
For example, the equation of the fitted line is , where is the height of the dog and is the predicted weight of the dog.
The slope is 4.27, which tells us that for every 1-inch increase in dog height, the weight is predicted to increase by 4.27 pounds.
In our example of the fuel efficiency and weight of a car, the slope of the fitted line shown is -0.01.
This tells us that for every 1-kilogram increase in the weight of the car, the fuel efficiency is predicted to decrease by 0.01 mile per gallon (or, after multiplying both values by 100, every 100-kilogram increase corresponds to a predicted decrease of 1 mpg).
When we have a negative association between two variables, an increase in one means there tends to be a decrease in the other.
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