## 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.

## 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.

## Decision Modeling.

Complete Problem 18 in Chapter 4 on page 159 about Decision Modeling;

A quality inspector picks a sample of 15 items at random from a manufacturing process known to produce 10% defective items.

Let X be the number of defective items found in the random sample of 15 items. Assume that the condition of each item is independent of that of each of the other items in the sample. The probability distribution of X is provided in the file P04_18.xlsx.

Use simulation to generate 500 values of this random variable X. Calculate the mean and standard deviation of the simulated values.

How do they compare to the mean and standard deviation of the given probability distribution? (Albright, 2017, p. 159).

In the discussion area, answer both questions in Parts a and b. Attach the Excel document

## Quadratic Equations and Functions

Remember the form of a quadratic equation: ax2+bx+c

You will use: W=-.01×2+100x+c where (-.01×2+100x) represents the store’s variable costs and c is the store’s fixed costs.
So, W is the stores total monthly costs based on the number of items sold, x.
Think about what the variable and fixed costs might be for your fictitious storefront business – and be creative. Start by choosing a fixed cost, c, between \$5,000 and \$10,000, according to the following class chart:

If your last name starts with the letter Choose a fixed cost between
S-T \$8600-\$9200
Your monthly cost is then, W = -.01×2+100x+c.
Substitute the c value chosen in the previous step to complete your unique equation predicting your monthly costs.
Next, choose two values of x (number of items sold) between 100 and 200. Again, try to choose different values from classmates.
Plug these values into your model for W and evaluate the monthly business costs given that sales volume

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