Notes, summaries, assignments, exams, and problems for Mathematics

Faculty and Student Responses

A university president proposed that all students must take a course in ethics as a requirement for graduation. The table below gives the responses of a sample of faculty and students at the university when asked their opinion on this issue.

Y: FAVOR O: Oppose N: Neutral

Totals

F: Faculty         45             15             10                    70

S: Student        90             110             30                  ₂₃₀ 

 Totals   135               125                40

Probabilities

1. Find the following probabilities; each time ONE person is being randomly selected.

P(S or Y) = (230 + 135 – 90)/... Continue reading "Calculating Probabilities from a Table: Ethics Course Requirement" »

Systematic Sampling: Type of Probability Sampling Method

In systematic sampling, sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. The sampling interval, calculated by dividing the population size by the desired sample size, determines the selection.

Oversampling: Techniques to Adjust Class Distribution

Oversampling is a data analysis technique used to adjust the class distribution of a data set, ensuring a balanced representation of different classes/categories.

P Value: Probability of Obtaining Extreme Test Results

The p value represents the probability of obtaining test results at least as extreme as the observed results during the test, assuming the null hypothesis... Continue reading "Understanding Systematic Sampling and Statistical Significance" »

Worksheet: Revisiting Calculating Probabilities from a Table

1. Goals

P(F) = 251/478 = 0.525

The probability that a randomly selected student is a girl is 0.525.

P(P^C) = (247 + 90)/478 = (478 – 141)/478 = 337/478 = 0.705

The probability that a randomly selected student did not indicate being popular as a primary goal is 0.705.

P(B or S) = (227 + 90 – 60)/478 = 257/478 = 0.538

The probability that a randomly selected student is a boy or indicated a primary goal is to excel at sports or both is 0.538.

P(B and S) = 60/478 = 0.126

The probability that a randomly selected student is a boy and indicated a primary goal is to excel at sports is 0.126.

P(S|B) = 60/227 =... Continue reading "Calculating Probabilities from a Table" »

Operating Segments & Disclosures

Revenue Test Criteria

Both external and intersegment revenue are included in the revenue test.

Profit & Loss (P&L) Test

The P&L test includes 10% of the higher absolute value of intersegment revenue and expenses.

Management's Business Evaluation

Management evaluates business results along product, customer, or geographic lines, not based on legal entity results.

Defining an Operating Segment

• Includes state
• Reviewed by the chief decision maker
• Discrete information available
• Product is the primary breakdown, with geographic and customer as secondary.

Operating Segment Disclosure Rules

• Must reconcile to GAAP.
• For immaterial operating segments, revenue, P&L, and assets must be disclosed.

Non-Operating Segment

In the examples C = {1,2,3,4} and D = {3,4,5}

SymbolMeaningExample{ }Set: a collection of elements{1,2,3,4}A ∪ BUnion: in A or B (or both)C ∪ D = {1,2,3,4,5}A ∩ BIntersection: in both A and BC ∩ D = {3,4}A ⊆ BSubset: A has some (or all) elements of B{3,4,5} ⊆ DA ⊂ BProper Subset: A has some elements of B{3,5} ⊂ DA ⊄ BNot a Subset: A is not a subset of B{1,6} ⊄ CA ⊇ BSuperset: A has same elements as B, or more{1,2,3} ⊇ {1,2,3}A ⊃ BProper Superset: A has B's elements and more{1,2,3,4} ⊃ {1,2,3}A ⊅ BNot a Superset: A is not a superset of B{1,2,6} ⊅ {1,9}A^cComplement: elements not in AD^c = {1,2,6,7}
When  = {1,2,3,4,5,6,7}A − BDifference: in A but not in B{1,2,3,4} − {3,4} =

Inventory Costing Methods: Situational Analysis

For each of the following situations, determine whether FIFO, LIFO, or Weighted Average would apply:

• FIFO would produce the highest amount of net income in an inflationary environment.
• FIFO would produce the highest amount of assets in an inflationary environment.
• FIFO would produce the lowest amount of net income in a deflationary environment.
• Weighted Average would produce the same unit cost for assets and cost of goods sold in an inflationary environment.
• LIFO would produce the lowest amount of net income in an inflationary environment.
• Weighted Average would produce an asset value that was the same regardless of whether the environment was inflationary or deflationary.
• LIFO would produce the lowest

Chapter 3: Key Concepts

Missing Variable

x = given Median * 2 - ((1 of 2) median numbers if even): ascending order

Unreadable Score

(U * N of unreadable) - (U * N of readable)

Degrees of Freedom

1 # of freedom is lost

Chebyshev's Theorem

K = Deviation

Calculate time % within K Dev of mean - (1 - 1/k^2) * 100 = %

Finding Values

Find x K devs from mean - (mean - K * given standard dev) = XLow...(mean + k * given standard dev) = XHigh

Finding % Within Times Xlow-Xhigh

K = (XHigh - Mean) / Given Standard dev....then do Chebyshev's - (1 - 1/k^2) * 100 = %

Z-Score Simplified

(Mean - Given Mean) / Given Standard Dev = ZScore

Finding Minimum Score Accepted

(Min devs) * (Given Devs) + (Mean) = Min Score

Weighted Mean

Chart Xi / Wi /XiWi

Use Empirical Rule if distribution

Simple Random Sample

A sample designed in such a way as to ensure that (1) every member of a population has an equal chance of being included in the sample, and (2) every sample of size n has the same chance of occurring.

Selecting Elements

Subjects are selected at fixed sampling intervals by selecting every kth element thereafter. The interval between selections is determined by the ratio of the population size to the sample size (N/n). For example, if the population size is N=1,000 and a sample size of n=100 is desired, then the sampling interval is 1,000/100 = 10, so every 10th person is selected into the sample.

Different stages of Spain in the XIX