Do Now 18 – Balance Point Directions: Use the data table to answer

Transcription

Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Do Now 18
Name:
Date:
Do Now 18 – Balance Point Directions: Use the data table to answer the questions. 1. Calculate the balance point. 2. Explain whether it is reasonable to fit a line to the data. 3. The data is plotted as a scatterplot below. Draw a line that contains your balance point and seems to be a good approximation of the data. 4. Find the equation of the line you plotted in 3 using point-‐
slope form of a line. 12
10
8
6
4
2
5
10
Math 2 Week 7 Packet Page 8
Name:
Date:
Linear Regression on the Calculator – Classwork and Homework
We have been finding lines of best fit for quantitative data. Now, we will use our calculators to
make the process easier.
1. This table shows the average number of gallons
of milk a family drinks per week.
a. Follow directions on first page to find the
equation of the best-fit line. Write down
the equation.
family size
1
2
3
4
5
6
Name:
Date:
number of gallons of
milk
1.0
1.5
2.2
3.8
4.7
5.0
b. What is the y-intercept of this line? Explain what it represents about the
situation. Does it make sense?
c. What is the slope of this line? Explain what it represents about the situation.
Does it make sense?
d. Sally has eight people in her family. How many gallons of milk do you think they
drink in a week? Does your answer make sense? If Sally says her family drank 8
gallons of milk last week, do you believe her?
e. You run into your friend John at the store and he is buying 3 gallons of milk for
his family this week. How many people do you think are in his family? Does your
answer make sense? If John says his family has 4 people in it, do you believe
him?
Name:
Date:
2. Use your calculator to find the best-fit line equation for the data in the table.
a. Write down the equation.
x
–3
–1
0
1
3
y
4
3
2
1
0
b. Graph the data points and the best-fit line together on your calculator screen. (If
you don’t see five points and a line, try pressing [ZOOM][9] to fix.) Draw your
calculator screen below.
3. The table shows ages and heights for several maple
trees. Let x = tree age (in years), y = tree height (in feet).
a. Enter the data table into your calculator, and
find the best-fit line equation:
b. Use the equation to predict the height of a 60year-old tree. Does this make sense?
tree age
tree height
3
6
72
60
25
21
24
18
5
9
42
35
17
14
50
40
5
11
87
70
c. Use the equation to estimate the age of a tree that is 100 feet tall. Does this make
sense?
Name:
Date:
4. This table shows the amount spent on TV advertising in the USA in various years.
Year
1995
1997
1999
2001
TV advertising expenditures
32,720
34,593
36,830
40,140
(in millions of dollars)
a. Enter the data table into your calculator, and find the best-fit line equation:
b. Use the equation to predict the amount spent on advertising in 1996. Does this
make sense?
c. Use the equation to predict in what year the money spent on advertising will
reach 60,000 million dollars. Does this make sense?
5. The data table shows the number of farmers and the
number of cows at each of ten different farms. This
problem is about how these two numbers are related.
Let x = the number of farmers, y = the number of cows.
a. Find the best-fit line equation. You may round off the
numbers to three decimal places.
b. Predict how many cows there would be at a farm with
6 farmers. Show work.
number of
farmers
number of cows
1
8
10
4
2
1
4
3
2
12
3
48
73
27
14
8
19
17
11
80
c. Predict how many farmers there would be at a farm with 100 cows. Show work.
Name:
Date:
Correlation: Examining Residuals and Correlation Coefficient
Today, we will be more closely examining best-fit lines. Begin by creating a scatterplot for the
data given in the Excel file on the web site. Remember to label your axes appropriately.
Lately, we have been using our calculators to find equations of lines of best fit. Excel can
calculate best-fit lines (trendlines) as well. Now, we are going to examine part of the procedure
that the calculator and the computer use to find them. Excel can make this process easier for
us.
Residuals are the:
Car Crashes # Deaths (per every 100,000 people) y = 0.9515x -‐ 1814.1 R² = 0.76471 Year Math 2 Week 7 Packet Page 13
Name:
Date:
The goal of finding the best-fit line is to make the numbers in the residual column as small as
possible. Since some sets of data can contain hundreds of points, you need a single number
that represents the error for the entire data set. Adding the individual residuals does not really
help. One prediction that is 10,0000 too high offsets a prediction that is 10,000 too low.
A line of best fit is the graph of the linear equation that shows the relationship between two
sets of data most accurately (with the least errors).
When working with a line of best fit we want to assess how well the line is fitting the data. The
sum of square residuals (and thus positive numbers) is one way to measure, but the minimum
value that this can reach changes depending on the data. Instead, we use what is called the
linear correlation coefficient (r), which is a number that is always between –1 and 1.
Note: When calculating r, the linear correlation coefficient, the calculator uses the sum of
square residuals. They are related, but we are not going to go into how.
The number r, called the correlation coefficient, is a measurement of how closely the bestfit line fits the data.
• A value of r = 1 or –1 would say the line fits the data perfectly (perfect correlation).
• A value of r that is greater than 0.8 is considered strong correlation
• A value of r that is less than 0.5 is often considered weak correlation
• A value of r = 0 would stand for the worst possible fit (no correlation)
o
A positive value of r would mean:
o
A negative value of r would mean:
r=1
r = 0.99
r = 0.95
r = 0. 90
r = 0.70
r = 0.23
Our calculator will tell you r for any best-fit line. To turn this feature on:
Name:
Date:
Enter your original Excel data into your calculator and find the line of best fit. You will now
notice that the calculator displays r along with the equation.
Equation:
What value of r does your calculator display? What does this number tell you?
The calculator also display r2, the coefficient of determination. This allows you to determine
how certain one can be in making predictions from the best-fit line. This is the number that
Excel gives you. You can find r by taking the square root of r2.
Now you try it with some data about the weather and the number of visitors to a water park:
What is the best-fit line that Excel gives you?
Visitors to a Water Park
What is the R2 value that Excel gives you?
What is the value of r?
What would you say about the correlation of
the best-fit line to the data?
Temperature
(F)
# of Visitors
(thousands)
62
36.8
67
40.9
69
36.9
70
37.1
72
44.3
77
42.3
79
45.8
80
45.4
83
48.4
85
51.5
88
57.7
94
57.3
96
56.1
98
58.9
Predicted
Residuals
Homework
1. The table of data comes from an experiment by Tor Carlson in 1913 on
Saccharomyces cerevisiae (a type of yeast). The data show the number
of hours elapsed and the number of yeast cells per square unit of area
in a Petri dish.
a. Find the balance point of the data and add it to the table.
b. Make a scatterplot of this data below.
Name:
Date:
Hours
Yeast
Density
0
9.6
1
18.3
2
29.0
3
47.2
4
71.1
5
119.1
6
174.6
7
257.3
c. Using your calculator, find the line of best fit. Write the equation and r-value below,
and plot the best-fit line on your scatterplot.
d. With a different color pen or highlighter, mark the residuals on your scatterplot.
e. Either in Excel, with your calculator or by hand calculate all of the residuals and label
them in your scatterplot. You can either fill out the table below, or print out your Excel
work and paste it below.
Predicted
Residuals
Hours
Yeast Density
Density
0
9.6
1
18.3
2
29.0
3
47.2
4
71.1
5
119.1
6
174.6
7
257.3
Name:
Date:
f. Look at your r-value. What does this tell you about your best-fit line? (Make sure you
address the sign of your r-value and the magnitude of your r-value.)
2. This table gives the mean height in centimeters of boys ages 5 to 13 in the
United States. (Source: National Center for Health Statistics)
a. Using your calculator find the best-fit line.
a. What does the slope represent? Does this make sense?
b. What does the y-intercept represent? Does this make sense?
Age
Height (cm)
5
109.2
6
115.7
7
122.0
8
128.1
9
133.7
10
138.8
11
12
13
143.7
149.3
156.4
c. What does the data point (5, 109.2) represent? Does this make sense?
d. Predict the height of a 14-year old. Does this make sense?
e. Predict the height of a 30-year old. Does this make sense?
f. Predict how old a 130cm tall boy is.
g. What is the correlation coefficient, r, for your best-fit line?
h. What does r tell you about your best-fit line?
Name:
Date:
3. Explain how your calculator finds the line of best-fit using residuals.
4. For the scatterplots below estimate the r-value for the lines of best fit.
d.
h.
e.
g.
i.
Name:
Date:
Two-variable Data Test Review
You should be able to:
• Distinguish between categorical and quantitative data
• Construct two-way tables for categorical data
• Calculate joint frequencies, marginal distributions, and conditional distributions
• Construct segmented bar graphs
• Determine whether two categorical variables are associated
• Construct scatterplots for quantitative data
• Find lines of best fit for data that is approximately linear
• Determine the equation of a line in point-slope and slope-intercept forms
o Point-slope: y = m(x − x1 ) + y1
o Slope-intercept: y = mx + b
• Determine whether two quantitative variables are correlated
• Determine and interpret the correlation coefficient
€difference between correlation and causation
• Explain the
€ predictions based on best-fit lines
• Make appropriate
• Answer practical questions about data and justify your reasoning
1. The following is data collected on time spent on homework (min) and time spent
watching TV (min) each night.
Homework
1
4
5 4 4 2
5
2 6 5
5
3 3 3 4 2
4
11
(min)
0 0
5 6 6 3
7 8 5 8 2 8 8 9 5
7
Television
(min)
7
8
3
0
15
7
2
2
5
3
0
9
0
4
0
3
5
5
6
12
5
9
5
2
7
3
8
5
0
1
0
41
4
4
6
0
3
4
4
2
a. Construct a scatterplot for the data. USE GRAPH PAPER!
b. Explain your choices for which variable is on which axis and the scales of each
axis.
c. Draw in an appropriate line of best fit on your scatterplot.
d. Write the equation of the best-fit line in point-slope form.
e. Change into slope-intercept form.
f.
Name:
Date:
Find the line of best fit given by your calculator. How close is your line from part
e to the one given by the calculator?
g. What is the slope of the line from your calculator? Write a sentence explaining
what it represents about homework and watching television. Does this make
sense?
h. What is the y-intercept of the line from your calculator? Write a sentence
explaining what it represents about homework and watching television. Does
this make sense?
i.
Use your calculator to find the correlation coefficient for this data. Explain what
it tells you about the data.
Name:
Date:
2. The following chart shows data from a telephone survey of 16 American adults.
Respondent
Gender
Party Affiliation
Person #1
Female
Democrat
Person #2
Female
Democrat
Person #3
Female
Democrat
Person #4
Female
Republican
Person #5
Male
Republican
Person #6
Male
Democrat
Person #7
Female
Democrat
Person #8
Female
Republican
Person #9
Female
Republican
Person #10
Male
Democrat
Person #11
Male
Democrat
Person #12
Male
Republican
Person #13
Female
Republican
Person #14
Male
Democrat
Person #15
Male
Republican
Person #16
Male
Republican
a. Construct a two-way table for this data.
b. Choose one of the joint frequencies in your table. Write a complete sentence
explaining what it represents about this survey.
c. Choose one of the rows or columns and calculate the conditional distribution.
Write a complete sentence explaining what each value calculated represents
about this survey.
d. Determine whether gender and party affiliation are associated or independent.
Show your work by constructing segmented bar graphs. Explain your reasoning.
Name:
Date:
3. The following chart shows nutrition information for items at fast food restaurants.
a. Using your calculator, generate a scatterplot comparing total calories and
carbohydrates. Estimate the correlation coefficient for theses variables and
explain your reasoning.
b. Determine the equation for the line of best fit and the correlation coefficient
from your calculator. Write them below. How closely did your estimate match
the actual correlation coefficient?
c. A “Bacon Cheeseburger” from Five Guys has 920 calories. Use your line of best
fit to predict the grams of carbohydrates in this burger. Is this an appropriate
prediction?
Name:
Date:
d. An “Original Slider” from White Castle has 13 grams of carbohydrates. Use your
line of best fit to predict the total calories in this burger. Is this an appropriate
prediction?
4. A survey was conducted about whether a person was an organ donor and whether they
were married or single. The two-way table below summarize the data collected.
a. Fill in the missing boxes in the table.
b. What percentage of single people are not organ donors? Show or explain how
you got your answer.
c. What percentage of organ donors are married? Show or explain how you got
your answer.
d. What percentage of people surveyed are organ donors? Show or explain how you
got your answer.
e. Determine whether martial status and organ donor status are associated or
independent. Show your work by constructing segmented bar graphs. Explain
your reasoning.
5. Play the calculator game from class again. Did your performance improve? J

Do Now 18 – Balance Point Directions: Use the data table to answer

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