Well Ordering Principle; Mathematics

Well Ordering Principle.

Well Ordering Principle
Well Ordering Principle

Recall that an integer p is prime if p ≥ 2, and if a, b are positive integers such that p = ab then either a = 1 or b = 1.

Theorem. Every integer n ≥ 2 has a prime factor.
One way to prove this for a given integer n ≥ 2 is to apply the Well ordering

Principle to the set X = {d ∈ Z : d ≥ 2 ∧ d | n},

The set of all factors d of n such that d ≥ 2.

(a) Prove that X is not empty.
(b) Prove that if p is the minimal element of X, then p must be a prime number.                                                                                                                            (c) Finish the proof of the theorem.

Amount of Gasoline (Gallons)

Amount of Gasoline.

Amount of Gasoline
Amount of Gasoline

Let X Represent The Amount of Gasoline, In Gallons, Drivers Put In Their Cars When Fueling At The Small Downtown Gas Station.

The Model For This Variable X Is Uniformly Distributed Between 4 And 12 Gallons.

For this uniform distribution the mean amount of gas put in a car is 8 gallons and the standard deviation is about 2.3 gallons.

Your friend is looking at this model for the amount of gas and says:

“I remember some rule from my statistics class last year ~ something about there being a 68% probability that the amount of gasoline a randomly selected driver will put in the car is within one standard deviation of the mean. So would that work in this case?”

Your short answer to your friend is ‘No, it will not work in this case.”

Formulate your more complete answer to your friend that includes both finding the actual probability that the amount that a randomly selected driver will put in the car is within one standard deviation of the mean and an explanation as to why this is not consistent with the 68% value.

Goodness Of Fit Test; Statistics

Goodness Of Fit Test.

Goodness Of Fit Test
Goodness Of Fit Test

Use the Week 5 Data Set to create and calculate the following in Excel®:

Conduct a goodness of fit analysis which assesses orders of a specific item by size and items you received by size.

Conduct a hypothesis test with the objective of determining if there is a difference between what you ordered and what you received at the .05 level of significance.

Identify the null and alternative hypotheses.

Generate a scatter plot, the correlation coefficient, and the linear equation that evaluates whether a relationship exists between the number of times a customer visited the store in the past 6 months and the total amount of money the customer spent.

Set up a hypothesis test to evaluate the strength of the relationship between the two variables. Use a level of significance of .05.

Use the regression line formula to forecast how much a customer might spend on merchandise if that customer visited the store 13 times in a 6 month period.

Consider the average monthly sales of 2014, $1310, as your base to:
  1. Calculate indices for each month for the next two years.
  2. Graph a time series plot.
  3. In the Data Analysis Toolpak, use Excel’s Exponential Smoothing option.
  4. Apply a damping factor of .5, to your monthly sales data.
  5. Create a new time series graph that compares the original and the revised monthly sales data.

Digital Logic Design; Signed Multiplier

Digital Logic Design.

Digital Logic Design
Digital Logic Design

Signed Multiplier; Digital Logic Design

Create a 4 bit Signed Multiplier with the following specifications:


A; 4 bit 2’s complement binary number. This could be positive or negative.
B; 4 bit 2’s complement binary number. This could be positive or negative.


8 bit 2’s complement binary number (This could be a positive or negative number)

The overall circuit should look like this:

At a minimum, the circuit must be implemented using controlled inverts and an unsigned multiplier as discussed in class:


1) The Controlled Inverts and Unsigned multiplier must be implemented as sub-circuits as demonstrated in class. These sub-circuits are used to build the 2’s Complement Signed Multiplier.

2) You are only allowed to use the basic gates: AND, OR, XOR, NOT.

3) You are NOT allowed to use Logisim’s built in circuits (i.e. Adders and Multiplexers). However, you are allowed to use the basic gates to build your own Adders and Multiplexers.

4) Once completed, please ZIP your CIRC file in the following format: CS212_Lastname_Firstname.zip.