An ice cream company collected data on their ice cream cones sales over a month in July in a Chicago suburb, along with daily temperature and the weather. The company is interested to develop a correlation between ice cream sales to the hot weather. Market research showed that more people come out in certain neighborhoods, to either enjoy the nice weather, or venture out if they do not have air conditioning in their apartments. The Chicago Police also tracked crime statistics during the same period. Crime statistics included murder, assault, robbery, battery, burglary, theft and motor vehicle theft. The data are shown below:

1.)Develop a linear regression model for ice cream sales over daily temperature. Show the linear equation in the form of y = ax + b, and the coefficient of determination.

What would be the projected forecast of ice cream sales in units, for daily temperature of 94 F?

2.) On July 15 & 16 there were heavy down pour of rain, which might have prevented some to venture out to purchase ice cream during the day. If you were to override those 2 data points, what would be the linear regression model be (by deleting July 15 & 16 data).

which would be considered a better forecast for ice cream sales

3.) Develop a linear regression on ice cream sales to crime statistics. Show the linear equation in the form of y = ax + b, and the r-square value.

Does this correlation demonstrate causation, that high ice cream sales cause crime statistics to go up?

Solve using the Laplace transform: y" + 4y = g(t) where y(0) = y'(0).

Hint: Use the convolution theorem to write your answer. You may leave your answer expressed in terms of an integral.

A model for the population

P(t)

in a suburb of a large city is given by the initial-value problem

= P(10⁻¹ − 10⁻⁷P),    P(0) = 3000,

where t is measured in months. What is the limiting value of the population?


At what time will the population be equal to one-half of this limiting value? (Round your answer to one decimal place.)
months

In order to apply Green’s theorem, the line integral of the boundary should be evaluated such that the integration region inside the boundary lies always on the left as one advances in the direction of integration. What happens if the region lies on the right? How can you apply the theorem then? Explain.

The centers of two circles are 4 cm apart, with one circle having a radius of 3 cm and the other a radius of 2 cm. Find the area of their intersection.

PART 2: Instructions: Write three paragraphs to answer this question:

Suppose a sociologist at John Jay College is given a grant to study “whether there is a binge drinking problem on the campus (and in the dorms and nearby neighborhood) of Small Town College (STC). Describe three different data collection methods that the John Jay sociologist could use to gather evidence to determine whether or not STC has a serious “binge drinking problem” on its campus.For each data collection method, suggest the questions or issues that the sociologist would focus upon.

Prove that (ZxZ, *) where (a,b)*(a',b') = (a+a',b+b') is a group

Information theory

Consider a random variable representing coin throws (Bernoulli Variable with Σ = {0,1} ). Let the true
probability distribution be p(0) = r, p(1) = 1-r.
Someone guesses a different distribution q(0) = s, q(1) = 1-s.
(a) Find expressions for the Kullback–Leibler distances D(p||q) and D(q||p) between the
two distributions in terms of r and s.
(b) Show that in general, D(p||q) ≠ D(q||p) and that equality occurs iff r = s.
(c) Compute D(p||q) and D(q||p) for the case r = 1/2 and s = 1/4.

Consider the curve traced out by the parametric equations: { x = 1 + cos(t) y = t + sin(t) for 0 ≤ t ≤ 4π.

1. Show that that dy dx = − 1 + cos(t)/sin(t) = − csc(t) − cot(t).

2. Make a Sign Diagram for dy dx to find the intervals of t over which C is increasing or decreasing.

• C is increasing on: • C is decreasing on:

3. Show that d2y/dx2 = − csc2 (t)(csc(t) + cot(t))

4. Make a Sign Diagram for d 2 y dx2 to find the intervals of t over which C is concave up or concave down.

• C is concave up on: • C is concave down on:

5. Using all of your work from numbers 1 through 4, sketch a detailed graph of C. Label the points (x, y), if any, where C has horizontal tangents, vertical tangents, or where C is not smooth.

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specific value of n. (Round your answer to six decimal places).  
∫₁³ sin (?) / ? ?? , ? = 4
Please show all work.

You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find out how much the account needs to hold to make this possible. Round your answer to the nearest dollar.

regular withdraw: $1900
interest rate: 2%
frequency: quarterly
time: 28 years

Given y 1 ( t ) = t2 and y2 ( t ) = t ^− 1 satisfy the corresponding homogeneous equation of

t^2 y ' ' − 2 y = − 3 − t , t > 0

Then the general solution to the non-homogeneous equation can be written as y ( t ) = c1y1(t)+c2y2(t)+yp(t)

Use variation of parameters to find y p ( t ) .

Discrete Structures

Use a proof by contraposition to show if x³ + 3x is an irrational number then so is x, for any real number x.