Section 6.4 : Scatter Plots and Trend Lines

Learning Targets: S.ID.6.c, S.ID.6, S.ID.6.a, S.ID.7, S.ID.8, S.ID.9

Important Terms and Definitions

Scatter Plot: a graph that relates two groups of data. To make a scatter plot, plot the two groups of data as ordered pairs. Most scatter plots are in the first quadrant of a coordinate plane, because the data are usually positive numbers.

Positive Correlation: Both sets of data increase together

Negative Correlation: one set of data decreases as the other increases

No Correlation: data sets are not related

Trend Line: Shows a correlation more clearly

• Interpolation: Using a trend line to estimate a value between two known values • Extrapolation: Using a trend line to predict a value outside the range of known values

Line of Best Fit: The trend line that shows the relationship between the two sets of data most accurately. This can be found using your graphing calculator!

• When calculating the line of best fit, your calculator will also tell you how closely the data is clustered around the line. It does this through the correlation coefficient (r). This is a number between −1 and 1. The closer the number is to −1 or 1, the closer to the line that data points lie. If r is close to 1, the line of best fit has a positive slope. If r is close to −1, the line of best fit has a negative slope.

Causation: When a change in one quantity causes a change in another quantity. Correlation does not necessarily imply causation.

Note: Sometimes a linear function isn’t the best representation of a scatter plot. Your calculator can also find exponential models in those situations.

Making a Scatter Plot

Simply treat each data set as an (𝑥, 𝑦) coordinate and plot the points!

(ex 1) The table below shows the number of hours worked and the amount of money each person earned. Make a scatter plot of the data. Is there a positive correlation, negative correlation, or no correlation?

Name Hours worked

Amount Earned

Ricky 6 $25.50 Snooki 12 $51.00 Lauren 11 $46.75 Derrick 9 $38.25 Ben 15 $63.75 J.D. 10 $42.50

Identifying Correlation

(ex 2) Suppose you take a survey of all the schools in your state. What would you expect the relationship between the number of students and the number of teachers in each school to be? Explain.

(ex 3) Consider the relationship between the number of times a car stops to fill its gas tank and the amount of gas the tank can hold. Would this have a positive correlation, negative correlation, or no correlation?

Body Length of a Panda

Age (months) 1 2 3 4 5 6 8 9 Body Length (in) 8.0 11.75 15.5 16.7 20.1 22.2 26.5 29.0

Find the Equation of a Trend Line

Step One: Create a scatter plot

Step Two: Draw a line that goes through/is close to as many of the points as possible.

Step Three: Pick two of the points on the line, and use them to find the slope and write an equation.

(ex 4) Make a scatter plot of the data below and draw a trend line. Find an equation for the trend line. Estimate the number of Calories in a fast-food that has 14 grams of fat.

(ex 5) Make a scatter plot of the data below and draw a trend line. Find an equation for the trend line. Estimate the body length of a 7 month old panda. Could you also use this data to extrapolate the length of a 3-year old panda? Explain.

Causation

(ex 6) Consider the relationship between the amount of time you spend exercising and the number of calories you burn. Is there likely to be a correlation? If so, does the correlation reflect a causal relationship? Explain.

(ex 7) Consider the relationship between a person’s height and the number of letters in the person’s name. Is there likely to be a correlation? If so, does the correlation reflect a causal relationship? Explain.

Using Your Graphing Calculator

See handout given in class

Homework – Section 6.4 : Scatter Plots and Trend Lines

Make a scatter plot of the data shown in each table. Describe the type of correlation the scatter plot shows.

1. Sweater Sales Average

Price $21 $28 $36 $40

Number Sold

130 112 82 65

2. Gasoline Purchases Dollars Spent

10 11 9 10 13 5 8 4

Gallons Bought

2.8 3.2 2.6 2.7 4.0 1.5 2.3 1.3

Consider each of the following situations. Would you expect a positive correlation, negative correlation, or no correlation between the two data sets? Explain.

3. weight of a baby at birth and the month in which the baby was born 4. the sale of snow blowers and the amount of snowfall 5. the amount of free time you have and the number of classes you take

6. Describe three situations: one that shows positive correlation, one that shows negative

correlation, and one that shows no correlation.

Make a scatter plot of the data below and draw a trend line. Find an equation for the trend line.

7. Length and Wingspan of Hawks Length (in.)

21 21 18 24 16 19 17 19

Wingspan (in.)

36 41 38 46 31 39 35 46

Estimate the wingspan of a hawk that is 28 inches long

8. Memory Test Study Time (min)

1 2 3 1 1.5 4 5 4 3 3.5

Response Time (s)

75 74 62 85 70 38 22 25 40 51

Estimate the response time for someone that studied for 2.5 minutes

Consider each of the following situations. Is there likely to be a correlation? If so, does the correlation reflect a causal relationship? Explain.

9. The amount of time you spend studying for a test and the score you earn 10. The number of pounds a student can bench-press and his/her chances of making the

football team 11. The number of cigarettes a person smokes each day and his/her chances of getting lung

cancer

Use your graphing calculator to find the equation of the trend line and the correlation coefficient. If a linear function does not give the best representation of the data, use an exponential model.

12. Wind Chill Temperatures for 15 mph winds Air Temp (℉)

35 30 25 20 15 10 5 0

Wind-Chill Temp (℉)

16 9 2 −5 −11 −18 −25 −31

Predict the wind-chill temperature when it is −5 ℉

13. Spartan Corp. Months 1 2 3 4 5 6 Stock Value (in dollars)

4.10 5.40 7.30 9.93 13.51 17.90

Predict the value of the stock after 10 months

14. Bacteria Culture Hours 1 2 3 4 5 6 7 8 # of Bacteria 1200 780 510 330 210 140 100 85

Predict the number of bacteria after 12 hours