How can we use mathematics in real life?

Assume that you won $300,000 and you want to spend half of the money and invest the rest (or
invest all). You need to make a decision on how to invest the money and you need to persuade
your partner (or parents) that you made the right choice.
A. List two or three financial institutions and their terms and conditions (interest rate, how
the interest is calculated, minimum balance, etc…)
B. Calculate what your savings would be in 10 years (or 20 years) in each institution that
you listed. Justify your answer (show your work).
C. What financial institution would you choose? And why?
D. Discuss the conclusion you draw from your calculations, and reflect how the mathematics
helped you make the decision.

Sample Solution

Mathematics

One of the beauties of mathematics is that it is able to provide help in all sorts of different situations and to all sorts of people.

You have land that you would like to use to create two distinct fenced-in areas in the shape given below. You have 410 meters of fencing materials to use. What values of x and y would result in the maximum area that you can enclose?

Sample Solution

ALGEBRA 1 SOL STATE TEST

Basically, the focus group and interviews all had a commonality in that technology helps ELL students and
Mathspace is a good tool. Saw improved scores and student engagement using MathSpace. I have attached
raw data of 31 students with test scores for the 6 units of algebra 1. Please discuss units and test scores.
Included student demographics with average grade level of 11 and an ESOL language level of 2 or a scale of
1-6 with 1 being lowest and 6 fluent. Also average scores for all units and the end of class SOL test is included.
Please discuss trends, etc.
The findings are that technology does help ELLs and MathSpace in this instance showed increased scores,
better math understanding and engagement in ELLs and the Algebra 1 pass rate improved. Drawbacks are
that there was a slight learning curve for students to learn the program. Again, please use sections 1 and 2 as
a guide to write section 3.0 and 4.0 as everything needs to tie back to each other.
Sections 1 and 2 of my paper have been approved and does not need any changes. Please fix parts in
sections 3 and 4 with professor’s comments. Please leave track changes on so that I can easily see the
changes made. Use the sample project attached as a guide to see the writing style and details needed.

Sample Solution

Equation Solving

Problem 1 (10 points)
Let z be a 2 ◊ 1 random vector with distribution,
z ≥ N
350
0
6
,
5 1 ≠1
2
≠1
2 1
64 .
Find a 2 ◊ 2 matrix A such that ÎAzÎ2 has a chi-square distribution, and specify the associated degrees of
freedom.
Problem 2 (15 points) 3+3+3+6 = 15
Consider the following two regression models (at the population level):
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4×2
2 + —5x1x2 + Á.
Assume that we fit the two models to the same data.
(a) Can we compare the R2 of the two models without looking at the results of the fittings? Justify your
answer.
(b) Can we say that the residual vector of one model is orthogonal to the fitted value vector of the other
model? If so specify the details, if not argue why not.
(c) Assume that the data is actually generated from the following model (at the population level)
y = —ú
0 + —ú
1×1 + —ú
2×2 + —ú
3x1x2 + Á,
with —ú
0 , —ú
1 , —ú
2 and —ú
3 all nonzero. Which of the two models M1 and M2 will produce an unbiased
LSE of —ú? Justify your answer.
(d) Now assume the data of sample size n = 100 is actually generated from model M1 with noise vector
following our standard Gaussian assumption Á ≥ N(0, ‡2In). Consider an F-test for comparing M1
and M2. What is the probability that the p-value of that test is Ø 0.05? Justify your answer.
1
Problem 3 (20 points) 10+10 = 20
Consider the following two regression models (at the population level):
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4x1x2 + Á.
We fit the two models to the following data (n = 10):
t(dat)

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

X1 -0.4 0.6 -0.2 0.8 0.9 -0.9 0.1 0.8 0.1 -0.1

X2 0.9 -0.1 0.4 0.1 -0.8 0.8 -0.5 -0.9 -0.3 0.9

y 2.0 -0.3 0.7 -0.4 -0.6 2.0 -0.4 -1.0 -0.2 0.9

(a) Compare the two models and choose one based on an F-test at level – = 0.05.
(b) Find the variance inflation factor for —1 in model M2.
Problem 4 (20 points)
Consider the following three regression models (at the population level):
M0 : y = —0 + Á,
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4×2
2 + —5x1x2 + Á.
We fit the three models to the same data with sample size n. For i = 0, 1, 2, let SSE(Mi) œ R and µˆ(Mi) œ Rn
denote the SSE and the fitted value vector of model Mi, respectively. The following information is given:
• The sample size n = 15.
• εˆ(M0) ≠ µˆ(M1)Î2 = 4 ◊ SSE(M1).
• The F-statistic for an F-test comparing models M1 and M2 is Fˆ = 4.
What is the R2 for model M2?
2
Problem 5 (15 points) 3 + 3 + 3 + 3 + 3 = 15
Consider regression model y = —0 + q4
j=1 —jxj + Á with the data given by
t(dat)

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

X1 0.0 -1 0.0 -1 -1 0 1.0 -1.0 1 0

X2 -1.0 0 1.0 1 0 -1 0.0 0.0 0 -1

X3 -1.0 1 1.0 -1 1 1 1.0 1.0 0 0

X4 -1.0 -1 1.0 -1 0 0 -1.0 0.0 0 0

y -0.7 -1 -0.1 1 -1 -4 -0.8 -0.3 -3 2

Under the standard assumption Á ≥ N(0, ‡2In), answer the following:
(a) Which of the 10 data points has the highest influence on the regression, based on the Cook’s distance?
(b) Which of the 10 data points has the highest leverage without having much of an influence on the
regression?
(c) Using the PRESS statistic, among the the following two models, which one should be selected:
(1) the model that includes x1 and x2 or
(2) the model that includes x3 and x4?
Both models also include the intercept.
(d) Is it reasonable to use R2 to select among the two models in part (c)? Justify your answer.
(e) Argue that the comparison based on R2 in part (d) is equivalent to comparing the p-values of two
F-tests. Specify those tests.
Problem 6 (20 points) 10+10 = 20
Consider linear regression model y = —0 + q2
j=1 —jxj + Á with the transposed design matrix XT given by

[,1] [,2] [,3] [,4] [,5]

X1 0 -1 0 -1 -1

X2 0 1 -1 1 0

and the transposed response vector yT given by

[,1] [,2] [,3] [,4] [,5]

y 3 -1 -4 -5 -3

The noise vector Á = [Á1,…, Án]
T follows the following model: (n = 5)
Á1 = z1
Ái ≠ Ái≠1 = zi, i = 2,…,n
zi ≥ N!
0, i
n

, i = 1, . . . , n.
Assume that {z1, z2,…,zn} are independent.
(a) What is the distribution of Á = [Á1, Á2,…, Án]
T ?
(b) What is the smallest possible variance of an unbiased estimate of —2?

Sample Solution

The graphs of polar equations

XP 17 M152 / SP2021 / Bingham

Directions: Answer the questions posed below, submit your answers to CANVAS as a Written Submission (direct written entry into CANVAS, using CANVAS writing tools) or as a File Upload. This XP is worth a maximum of 10 points.

1a) The graphs of polar equations and have two points of intersection.
Identify those points using Polar coordinates AND rectangular coordinates.

1b) The graph of linear equation has two intercepts.
Identify those points using rectangular coordinates AND polar coordinates.
Then, write the original equation in Polar function form. (that’s form)

2a) Using your calculator/technology to help, sketch the graph of the Polar function
on a set of Polar axes.

Graph it over the interval ; mark the orientation of the graph.

Also, mark the positions and give the exact Polar coordinates of the points where

 .       (hint:  shows up a lot)

b) Find the Polar radial derivative for the function in part a. Then, use it to find the exact values of for the values mentioned in part a, above.
At which two points in part a’s graph does = 0?

Sample Solution

Mathematics

Determine the value of each of the following limits (if they exist).
(8pt) Use the method of your choice, be sure to show all work. If you are using a theorem from your textbook,
then you must state it by name.
a. 𝑙𝑖𝑚
𝑥→∞
𝑥
2−3𝑥+2
3𝑥−8
b. 𝑙𝑖𝑚
𝑥→2
𝑥
2−3𝑥+2
2𝑥
2−8

Sample Solution

DESIGNING A MATH LESSON

Select a grade level K-3 and at least one state standards related to mathematical operations. Using the “COE Lesson Plan Template,” develop a lesson based on your selected standards.
K.MATH.9. [NY-K.CC.5a.] Answers counting questions using as many as 20 objects arranged in a line, a
rectangular array, and a circle; answers counting questions using as many as 10 objects in a scattered configuration*

As you are developing your lesson, consider how to create objectives that measure students’ actions and incorporate differentiated learning to meet the needs of students at, above, and below grade level.

What knowledge and skills would need to be taught before this lesson to make sure students are able to retain the content?
What lessons would logically be taught after this lesson to take students to the next level of understanding?
How would you differentiate to meet the needs of students above and below grade level?

Sample Solution

Trigonometric question

Suppose on a nice December evening you gaze upon your favorite star. You stick your handout into the sky and point at it, then you keep your arm fixed and frozen. Until 6 months later when you reach for the sky again and point at your star with your other arm. Suppose now you observe the angle you have created with your arms to be 0.2 degrees . How far is the star from the sun? [hint: assume the earth orbits in a circular motion around the sun once every 12 month, where the orbit has radius of 149.6 × 10 6km .

Sample Solution

Math Action Plan

The Math Action Plan assignment will outline a detailed plan on what math you want to study, how and when you plan to learn this math, and examples of the math concepts. Use the links in the Resources section of this course to assist you in this task

Sample Solution