Algebra 1 Questions

I have listed the questions and their explanations below. Please help me understand the concept and how to solve these questions. Thank you.

Match each correlation coefficient to the appropriate scatter plot. The line in each scatter plot is
the least squares regression line.
r = 0.1
r=-0.4
601
54
48
42
***
36
30
24.
18
601
54
48
42
36
30
24
18
12
6
12
6
0
6 12 18 24 30 36 42 48 54 60
ot & 12 18 24 30 36 42 48 54 60
You answered:
– 0.1
-0.4
6011
54
601
.
48
541.
487
42
36
30
.
42
36
24
30
24.
18
12+
6+
01 6 12 18 24 30 36 42 48 54 60
18
12
6
0 6 12 18 24 30 36 42 48 54 60
The correlation coefficient, r, measures how close a set of data points is to being linear. In other
words, it measures the strength of the linear correlation of the data set.
• Larger values of Ir mean the data set has a stronger linear correlation. Smaller values of
Ir mean the data set has a weaker linear correlation.
• The correlation coefficient, r, is between-1 and 1. It is 1 if the data points are on a line
with positive slope. It is -1 if the data points are on a line with negative slope.
• The correlation coefficient is positive if the data points have a positive (increasing) trend,
and negative if the data points have a negative (decreasing) trend. It has the same sign
as the slope of the least squares regression line.
This scatter plot shows a positive (increasing) trend.
a )
601
.
601
54
48
42
36
30
241
18
12
6
0 6 12 18 24 30 36 42 48 54 60
54
487
42
361
30+
24
18
12
6
24
0 12 18 24 36 36 42 48 54 609
6 60
Find the correlation coefficient, r, of the data described below.
The Patel toy company is testing the usability of its new spinning top among
different age groups. Dina, a designer at the company, gave the tops to a group
of children of different ages and had each child spin a top once.
Dina recorded the age of each child, x, and how long he or she got the top to spin
(in seconds), y.
Age Time (in seconds)
6
28
8
76
8
66
9
79
10
39
Round your answer to the nearest thousandth.
You answered:
The correlation coefficient, r, for a set of n data points, (Xi, Yi) where 1 sis n, measures
the strength of the linear correlation of the data set. The closer r is to 1 or -1, the stronger
the linear correlation of the data.
The correlation coefficient is defined by the formula:
1
(x – x)(y-7)
n-1
S.S,
where:
• x is the mean of the values xi
y is the mean of the values yi
• Sx is the sample standard deviation of the values xi
• Sy is the sample standard deviation of the values yi
.n is the number of data points
In practice, you can use a calculator or computer to find the correlation coefficient.
Each row in the table shows a data point (Xi, yi), where:
Xi = the age of a child
Vi = how long that child made the top spin (in seconds)
Use your calculator to find the correlation coefficient of the given data set:
r = 0.320079…
ដ 0.320
Round to the nearest thousandth
Find the equation for the least squares regression line of the data described below.
Brian wants to figure out how long it usually takes to get through a supermarket
checkout line. For several weeks, he observed the checkout lines he waited in.
Brian counted how many people were ahead of him in each line he joined, x, and
how many minutes it took him to get to the front of that line, y.
People Minutes
2
3
3
3
3
5
6
19
8
20
Round your answers to the nearest thousandth.
9 –
x +
You answered:
ŷ =
A line can be used to predict values based on a data set. You can look at residuals to see how
well a line fits the data. Residuals are the differences between the observed values and the
values predicted by the line.
2017
18
16
14
12
10
8
6
4+
2
0
2
4
6 8 10 12 14 16
The least squares regression line, ý = ax +b, is the line for which the sum of the squares
of the residuals is minimized. It is commonly used to make predictions for a data set.
In practice, you can use a calculator or computer to find the equation of the regression line.
Each row in the table shows a data point (xi, Yi), where:
xi = the number of people ahead of Brian in a line
Yi = how many minutes Brian had to wait in that line
Use your calculator to find the equation of the least squares regression line for the given data
set:
# = (3,333333… )x = 4.666666…
– 3.333x – 4.667
Round to the nearest thousandth
Read the following description of a data set.
Jonathan is packing for vacation and is wondering how many books to bring. He
wants to make sure he does not run out of reading material, so he looks at his
reading journal to see how many books he read on past vacations
For each vacation, he records the vacation’s length (in days), x, and how many
books he read, y
The least squares regression line of this data set is:
ý = 3.088x – 25.805
Complete the following sentence:
If Jonathan stays on vacation for one extra day, the least squares regression line predicts
that he would have read
additional books.
You answered:
If Jonathan stays on vacation for one extra day, the least squares regression line predicts that
he would have read
additional books.
A line can be used to predict values based on a data set. You can look at residuals to see how
well a line fits the data. Residuals are the differences between the observed values and the
values predicted by the line.
201
18
16
14
요
10
8
6
4
2
0
2
4
6
8
10 12 14 16
The least squares regression line, ý = ax +b, is the line for which the sum of the squares
of the residuals is minimized. It is commonly used to make predictions for a data set.
You can use the least squares regression line to predict the y-value for a given x-value. The
slope of the regression line is the predicted change in the y-value when the x-value increases
by 1.
So, the least squares regression line predicts the y-value will go up by 3.088 when x goes up
by 1. This means:
If Jonathan stays on vacation for one extra day, the least squares regression line predicts
that he would have read 3.088 additional books.
Find the equation for the least squares regression line of the data described below.
A team of forestry specialists from the Joseph lumber company visited a section of
the forest that was going to be logged for timber. They took measurements to
evaluate the health and growth of trees in the area.
The team counted the number of rings on several trees to find their ages (in years),
x. They also measured the diameter of these trees (in meters), y.
Age (in years) Diameter (in meters)
25
0.6
38
2.2
41
1.7
75
1.7
89
2.8
Round your answers to the nearest thousandth.
+
You answered:
ỹ =
A line can be used to predict values based on a data set. You can look at residuals to see how
well a line fits the data. Residuals are the differences between the observed values and the
values predicted by the line.
2017
18
16
14
12
10
8
6
4
27
0
2
6
8
10 12 14 16
The least squares regression line, ý = ax + b, is the line for which the sum of the squares
of the residuals is minimized. It is commonly used to make predictions for a data set.
In practice, you can use a calculator or computer to find the equation of the regression line.
Each row in the table shows a data point (xi, vi), where:
xi = the age of a tree (in years)
Yi = the diameter of that tree (in meters)
Use your calculator to find the equation of the least squares regression line for the given data
set:
ý = (0,021356…)x + 0.655294…
= 0.021x + 0.655
Round to the nearest thousandth