Math Tools: Chapters 5-8
Math Tools: Chapters 5 – 8
By Keegan Calligar
Chapter 5 – Polls and Surveys
Populations and Samples
- Aim for at least 400 interviews to ensure acceptable margin of error
- Census, universe or population sampling – sampling everyone in the population
- Cluster sampling – sampling one area or region, by zip code, county, etc.
- Multistage sampling – national samples; selecting specific area, then random subgroups, then individual blocks within sub-group, then smaller block
- Systematic random sampling – pick random number (n), then take phone book and calling every nth number
- Quote sampling – sample based on demographics
- Probability sampling – put all potential subjects in hat, draw out designated percentage
Margin of Error and Confidence Level
- Margin of Error – degree of accuracy of research based on standard norms
- Percentage, based on size of a randomly selected sample
- As the number of people polled increases, the margin of error decreases
- Confidence level – level / percentage at which researchers have in the results of their research
- Formal definition – probability of obtaining a given result by chance
- Predetermined
- Usually 90, 95 or 98 percent
- Should always be reported so that readers can analyze results for themselves
Census
- 2000 U.S. Census sent out through mail and by going door-to-door
- Results come in slowly and will continue for years
- Return rate of 2000 = 67 percent
- Adjusted figures – any figure that is statistically manipulated to compensate for missing data
- Census used for creating different, equally populated congressional districts
- Try to give smaller states slightly more representation
Z scores and t scores
- Frequently used to report results of studies
- Z score also called ‘standard score’ – shows how much a certain figure differs from the mean
- Standard deviation used as this unit measure
- Mean then becomes zero, first standard deviation is 1, second is 2, etc.
- t score, also called ‘student’s t distribution, closely related to z scores
- used when sample size is less than 100
- z score = (raw score – mean) / standard deviation
Chapter 6 – Business
Financial Statements
- Found in company’s annual report
- Include profit and loss report, balance sheet
Profit and Loss
- AKA P&L
- Shows if a company is making money
- To find, subtract expenses from income
- “Cost of goods sold” - direct expenses a business incurs in making or buying products
- Overhead – expenses the company would incur regardless of how many products it makes or sells … ex – insurance, employees, rent
- Gross margin – difference between cost of goods sold and selling price
- Net profit – how much a company makes after subtracting overhead from gross margin
- EBITDA = earnings before interest, taxes, depreciation and amortization – used to see how a company is doing before certain expenses
- can use to compare companies, because it shows how much cash each company is making before factoring in their different expenses
- aka “operation cash flow” – measures how much cash a given business has to operate with
- Formula: Gross Margin = Selling price – cost of goods sold
- Formula: Gross Profit = gross margin x number of items sold
- Formula: Net Profit = gross margin – overhead
Balance Sheet
- Written record of a given company’s assets, liabilities and equity
- Indicates company’s financial stability
- Formula: Assets = Liabilities + Equity
- Assets = resources owned by the company that are of some monetary value
- Current assets = cash, investments, liquid items of value
- Long-term assets = buildings, furniture, etc
- Accounts Receivable = money owed the company by customers
- Accumulated Depreciation = decline of the value of an asset
- Inventory = record of good on hand
- Intangible assets = copyrights, patents, and research with legal economic value
- Investments in other corporations = can be in stocks or influence in other company’s gain
- Fixed assets = property, plants, equipment, deferred charges
- Short-term investments = stocks, bonds
- Pre-paid expenses = rent, insurance
- Uncollectible accounts receivable = write-offs for bad debts, allowance for potential bad debts
- Equity – value of the company, owner’s and/or shareholders investments, capital accounts and other related assets
- Dividends – payments to shareholders that represent the distribution of the company’s assets
- Retained earnings = earnings set aside for future business purposes
- Liabilities – obligations that need to be paid later on
- Accounts payable – bills that need to be paid
- Accrued liabilities – liabilities that have occurred but are not yet paid
- Current or short-term liabilities – money owed to suppliers, interest on debt, taxes and wages
- Long-term liabilities = debt, deferred taxes, leases
Ratio Analysis
- Calculations used to analyze company’s cash, profitability, operating efficiency, and market value
- Show trends in company’s life
- Used to compare against similar companies
- Current ratio – liquidity ratio, measures ability of company to meet its liabilities
- Formula: Current ratio = current assets / current liabilities
- Quick ratio – liquidity ratio, measures ability of a company to meet its current liabilities with cash on hand
- Formula: Quick ratio = cash / current liabilities
- Debt-to-asset ratio – like current ratio, but includes all assets and all liabilities; good indicator of long-term health of a company
- Formula: Debt-to-asset ratio = total debt / total assets
- Debt-to-equity ratio – tells how deeply company is leveraged by comparing what is owned and what is owed
- Formula: Debt-to-equity = total debt / equity
- Return on Assets – profitability ratio; measures return on the investment on all assets
- Formula: return on assets = net income / total assets
- Return on Equity – profitability ratio; measures return on investment made in equity
- Formula: return on equity = net income / equity
- Price-earning ratio – value ratio; measures return of the investment based on stock price
- Formula: price-earnings = market price / share DIVIDED BY earnings/share
Chapter 7: Stocks and Bonds
Stocks
- Sold to raise cash, buy to invest
- Buy stock – become part owner of that company
- Mutual fund companies sell shares of funds, use that money to buy stock in other companies
- 52-Week Chart
- High/Low – highest and lowest stock prices in past year
- Stock – stock’s symbol
- Div – most recent annual dividend company paid to share holders, per share
- PE – Price/Earnings ratio
- Last – price of one share at the end of the previous day
- Change – how much stock value changed over one day
Bonds
- Bond is a loan from an investor to the government or other organization selling bonds
- Earn interest at a set rate, generally low-risk investments
- ‘Face Value’ – amount the owner will receive at maturity
- Current yield (return on investment) fluctuates because value of bond on open market flucates with supply and demand
- Formula: current yield = (interest rate x face value) / price
- Formula: Bond cost (interest) = amount x rate x years
Market Indexes
- Track prices of ceratin groups of stocks
- Dow Jones Industrial Average
- total value of one share each of 30 select socks divided by divisor
- divisor includes dividends, splits, spinoffs, etc
- able to provide snapshot of entire stock market
- thirty stocks represent about 1/5 of market value of all US stocks
- National Association of Securities Dealers Automated Quotations (NASDAQ)
- monitored by SEC
- automated quotation system
- reports on trading of domestic stocks and bonds not listed on the regular stock markets
- lists more than 5,000 securities
- member companies must be regiesterd with SEC, have at least two market makers, meet minimum requirements for assets, capital, public shares and stockholders
Chapter 8: Property Taxes
- Largest single source of income for local governments
- Determined by taking the total amount of money the governing body needs, and dividing that among the property owners in that taxing district
- Purpose of reappraisal is to update real property values to reflect current market value of all taxable properties within a taxing district
- Property is often taxed by more than one governing body
Mills
- 1/10 of a cent, or $0.001
- Formula: Mill levy = taxes to be collected by the government body / assessed valuation of all property in the taxing district
Appraisal Value
- Based on:
- Property’s use (residential, business, vacant, land, farm, commercial)
- Property’s characteristics: location, square footage, number of stories, exterior wall type, age, quality of construction, amenities
- Current market conditions – determined by sales in immediate area over given number of years
- Visual inspection of property by appraiser
Assessed value
- Percentage of market value
- Mill levy applied to assessed valuations
- Value depends on local policies
- Formula: Assessed value = Appraisal value x rate
Calculating Tax
- Formula: Tax owed = Tax rate x (assessed value of property / $100)
- Note: divide the assessed value by $1,000, rather than $100, if the rate is based on an amount per $1,000 of assessed value
Examples
Chapter 5
2. The Graduate Research Methods class decided to find out how many students on campus read a newspaper at least five times a week. Select one of the five sampling methods. Justify your selection.
To do this, I would use quota sampling, because, according to the book, it “aims to select the sample based on known demographic characteristics.” If you were trying to find out how many college students read the school newspaper, there would be no need to poll professors, staff, and others who live in the area but are not students. By selecting those who are in the known demographic – college students – the individual completing the survey insures more true answers.
Chapter 6
- The Town of Tuckahoe buys 1,000 tons of gravel from PK Wholesale Inc. at $10 a ton. Tuckahoe uses 670 tons to pave streets. The city manager wants to get rid of the excess gravel. He offers it to Gadsen for $13 a ton. Gadsen buys the gravel because a road collapsed and the city needs to make emergency repairs. What was Tuckahoe’s gross margin on the sale?
Gross margin = Selling price – cost of goods sold
Gross margin = (330 x 13) – (330×10)
Gross margin = 4,290 – 3,300
Gross margin = 990
Chapter 7
- Examine the newspaper stock table on page 102. Which stock’s price has fluctuated the most in the past year? Which stock has the highest price to earnings ratio? Which stock went up the most that day? Which went down the most that day?
- Aon’s stock flucated the most in the past year. it began at 44.80 and ended at 13.30, thus fluctuating 31.50 points.
- Apache had the highest PE ratio.
- Aon went up the most that day; it was the only stock to go up that day.
- Apache went down most that day, as it went down .63 points.
Chapter 8
- Rockham pegs its tax rate to 75 percent of true value. If a house at 5382 Brice Creek Road is appraised for $210,000, what is the assessed value of the house?
Assessed value = appraisal value x rate
Assessed value = 210,000 x .75
Assessed value = $157,500
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