Math 111 Final Exam Comprehensive Study Guide

Unit Conversions and Dimensional Analysis

• Vehicle Fuel Cost Calculations     - Calculations for travel expenses involve relating distance, fuel efficiency, and price per unit of fuel.     - Problem Scenario: A car with a fuel efficiency of $28\,\text{miles per gallon}$ plans a round trip to Pittsburgh ($80\,\text{miles}$ away) with gasoline priced at $\$3.79\,\text{per gallon}$.     - Total Distance: For a round trip, the distance is doubled. $80\,\text{miles} \times 2 = 160\,\text{miles}$.     - Fuel Requirement: The gallons needed are calculated by dividing total distance by miles per gallon: $\frac{160\,\text{miles}}{28\,\text{miles/gallon}}$.     - Total Cost: Calculated by multiplying the total gallons required by the cost per gallon.

• Time Management Calculations     - Estimating total time spent on activities requires converting hours to minutes over a set period.     - Problem Scenario: An average person spends $4.5\,\text{hours per day}$ on their phone.     - Monthly Projection: Assuming a standard month of $30\,\text{days}$, the total minutes are calculated as: $4.5\,\text{hours/day} \times 60\,\text{minutes/hour} \times 30\,\text{days}$.

• Currency and Metric Conversions     - International commerce requires converting mass (grams to kilograms) and currency (Euros to U.S. Dollars).     - Problem Scenario: Purchasing $250\,\text{grams}$ of chocolate in Belgium for $40\,\text{EUR\,per\,kg}$.     - Exchange Rate: $1\,\text{EUR} = \$1.08\,\text{USD}$.     - Calculation Path: Convert mass to kilograms ($0.250\,\text{kg}$), multiply by the price in Euros per kg, then apply the exchange rate to find the cost in U.S. Dollars.

• Average Speed in Multi-Stage Athletic Events     - Average speed is defined as total distance divided by total time: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$.     - Race Breakdown (2 miles total):         - First Half-Mile ($0.5\,\text{mile}$): Rate of $352\,\text{feet per minute}$ for $7.5\,\text{minutes}$.         - Second Half-Mile ($0.5\,\text{mile}$): Rate of $176\,\text{feet per minute}$ for $15\,\text{minutes}$.         - The second mile repeats the pace of the first mile.     - Distance Elements: Total distance is $2\,\text{miles}$ (or $10,560\,\text{feet}$ since $1\,\text{mile} = 5,280\,\text{feet}$).     - Time Elements: Total time is $(7.5 + 15) \times 2 = 45\,\text{minutes}$.

Index Numbers and Regional Cost Comparisons

• Housing Price Index (HPI)     - HPI is used to compare the relative cost of housing in different geographic locations.     - Reference Data (City Housing Indices):         - Dallas: $121$         - Miami: $161$         - Phoenix: $122$         - San Francisco: $172$     - Formula for Comparison: $\text{Price in City B} = \text{Price in City A} \times \frac{\text{Index of City B}}{\text{Index of City A}}$     - Phoenix to Miami Example: For a $\$400,000$ house in Phoenix, the Miami equivalent is found by multiplying $\$400,000$ by the ratio of Miami's index to Phoenix's index ($\frac{161}{122}$).     - Dallas to San Francisco Example: For a $\$300,000$ house in Dallas, the San Francisco equivalent is found by multiplying $\$300,000$ by the ratio of San Francisco's index to Dallas's index ($\frac{172}{121}$).

Statistical Analysis and Medical Treatment Outcomes

• Comparison of Medical Techniques for Kidney Stones     - Effectiveness is measured by the percentage of successful outcomes relative to total cases.     - Data Table:         - Treatment 1:             - Small Stones: $81\,\text{Successful}$, $6\,\text{Not Successful}$.             - Large Stones: $192\,\text{Successful}$, $71\,\text{Not Successful}$.         - Treatment 2:             - Small Stones: $234\,\text{Successful}$, $36\,\text{Not Successful}$.             - Large Stones: $55\,\text{Successful}$, $25\,\text{Not Successful}$.     - Analysis Tasks:         - Combined Success Rate for Small Stones: Sum successes from Treatment 1 and Treatment 2 for small stones, then divide by the total number of small stone cases (sum of all small stone successes and failures).         - Failure Rate for Treatment 1: Sum the "Not Successful" counts for both small and large stones under Treatment 1, then divide by the total number of Treatment 1 cases.         - Evaluation Choice: Determining the superior treatment for large stones requires comparing the success rate (Success / Total) specific to large stones for both Treatment 1 and Treatment 2.

Personal Finance: Interest, Mortgages, and Insurance

• Compound Interest and Savings Accounts     - Formula for Monthly Compounding: $A = P\left(1 + \frac{r}{n}\right)^{nt}$.     - Scenario Variables:         - Principal ($P$): $\$5,000$         - Annual Percentage Rate ($r$): $3.5\%$ (expressed as $0.035$)         - Compounding Periods per year ($n$): $12$ (monthly)         - Time in years ($t$): $7$     - Interest Earned: Calculated as $\text{Final Amount} (A) - \text{Initial Deposit} (P)$.

• Mortgage Comparison     - Standard home loans are typically evaluated by APR and loan term length.     - Loan 1: $30\text{-year}$ mortgage at $4.8\%\,APR$.     - Loan 2: $15\text{-year}$ mortgage at $4.2\%\,APR$.     - Key Comparisons:         - Monthly Payments: Shorter-term loans (15-year) generally have higher monthly payments than longer-term loans (30-year), even if the interest rate is lower.         - Total Interest: Longer-term loans (30-year) result in significantly higher total interest costs over the life of the loan due to the extended period of compounding.

• Insurance Policy Costs     - Premiums: The regular payment required to keep an insurance policy active.     - Deductibles: The fixed amount the insured must pay out-of-pocket for covered services before the insurance provider begins to pay.     - Co-payments: A fixed amount the insured pays for a specific service (e.g., a doctor's visit) after the deductible is met.     - Policy Evaluation Factors: Consider total cost (premiums + potential out-of-pocket), coverage limits, provider networks, and specific health or property needs.

Statistical Methodology and Study Design

• Classification of Data     - Qualitative (Categorical) Data: Represents attributes or categories. Example: The color of cars in a dealership parking lot.     - Quantitative (Numerical) Data: Represents measurable quantities or counts. Example: The number of SUVs in a car dealership parking lot.

• Observational Studies vs. Experiments     - Observational Study: Researchers observe and measure characteristics without attempting to modify the subjects being studied.         - Retrospective Study: An observational study where data is collected from the past (e.g., through records or interviews).         - Cases and Controls: In health studies, cases are individuals with a condition, and controls are those without.     - Experiment: Researchers apply a treatment to some subjects and observe the effects.         - Treatment Group: The group receiving the actual treatment (e.g., vaccine).         - Control Group: The group receiving a placebo or no treatment (to provide a baseline for comparison).         - Single-Blind: The subjects do not know if they are in the treatment or control group.         - Double-Blind: Neither the subjects nor the researchers interacting with them know who is in which group (minimizes researcher bias).

• Study Case Studies:     - Case A: A study of $2,002\,\text{runners}$ looking for relationships between injuries and physical variables (height, weight, age). Classified as an Observational Study.     - Case B: Measuring COVID-19 vaccine efficacy using $500\,\text{vaccinated subjects}$ and $500\,\text{placebo subjects}$. Classified as an Experiment. Blindness is necessary to prevent psychological effects or bias in reporting symptoms.     - Case C: A survey finding that $18.6\%$ of first-year women at a college experienced sexual assault. Classified as an Observational Study.

• Variable Identification     - Independent Variable: The factor being manipulated or controlled in a study (e.g., the amount of light given to plants).     - Dependent Variable: The outcome being measured (e.g., the height of the plants).     - Confounding Variables: Extra variables that were not accounted for that could affect the results (e.g., water amount, soil quality, temperature differences for the plants).

Data Visualization and Comparison Metrics

• U.S. Voter Affiliation Trends     - Analysis involves interpreting a graph tracking Republican, Democrat, and Independent affiliations over time.     - Absolute Difference: Calculated as $|\text{Value A} - \text{Value B}|$. Example: Difference between Republican and Independent shares in $2022$.     - Percent Difference: Calculated as $\frac{|\text{Value A} - \text{Value B}|}{\text{Average of A and B}} \times 100\%$.     - Estimation: Determining a value from a specific point on a graph (e.g., Independent share in $2004$).     - Percent Change: Calculated as $\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$. (e.g., Independent voter change from $2004$ to $2022$).

Descriptive Statistics and Probability

• Measures of Central Tendency and Spread     - Data Set (Cat Weights): $4$, $4$, $5$, $9$, $10$, $15$, $19$, $22$.     - Mean: Sum of all values divided by count ($n=8$).     - Median: The middle value (or average of the two middle values, since $n$ is even). Here, the average of $9$ and $10$.     - Mode: The value that appears most frequently (here, $4$).     - Range: Difference between the maximum and minimum values ($22 - 4 = 18$).

• Probability Calculations     - Scenario 1 (Vase of Flowers):         - Contents: $4\,\text{carnations}$, $2\,\text{roses}$, $8\,\text{daisies}$, $3\,\text{lilies}$, $7\,\text{tulips}$. Total flowers = $24$.         - Addition Rule (OR): Probability of selecting a rose or tulip = $\frac{\text{Roses} + \text{Tulips}}{\text{Total}} = \frac{2 + 7}{24}$.         - Multiplication Rule (AND - Dependent): Probability of picking 3 daisies in a row without replacement = $\frac{8}{24} \times \frac{7}{23} \times \frac{6}{22}$.     - Scenario 2 (Coffee Consumption):         - Data: $63\%\,(\text{or } 0.63)$ of Americans drink coffee every morning.         - Both Drink Coffee: $0.63 \times 0.63$.         - Neither Drink Coffee: $(1 - 0.63) \times (1 - 0.63) = 0.37 \times 0.37$.         - First Drinks, Second Does Not: $0.63 \times 0.37$.

Public Health Statistics and Growth Models

• COVID-19 Death Rates (2020 Data)     - Death Rate: Usually expressed as deaths per total population.     - Pennsylvania: $16,609\,\text{deaths}$ in a population of $12,989,625$.     - Regional Rate (Mid-Atlantic): Calculated by summing deaths in NY, NJ, PA, DE, and MD, then dividing by the sum of their populations.     - Regional Data Provided:         - New York: $35,736\,\text{deaths}$, $20,154,933\,\text{pop}$.         - New Jersey: $16,497\,\text{deaths}$, $9,279,743\,\text{pop}$.         - Pennsylvania: $16,609\,\text{deaths}$, $12,989,625\,\text{pop}$.         - Delaware: $1,008\,\text{deaths}$, $991,886\,\text{pop}$.         - Maryland: $6,000\,\text{deaths}$, $6,172,679\,\text{pop}$.

• Linear vs. Exponential Growth     - Context: Home Owner’s Association (HOA) dues increase.     - Linear Growth: Raising dues by a fixed amount ($\$5\,\text{every year}$).     - Exponential Growth: Raising dues by a percentage ($8\%\,\text{every year}$).     - Comparison: Starting with $\$65\,\text{per month}$, the linear increase results in a steady climb ($\$70$, $\$75$, $\$80$…), while exponential growth compounds ($65 \times 1.08$, then that result $\times 1.08$). In the long run, exponential growth will always exceed linear growth.

Social Choice and Voting Theory

• Voting Methods Analysis     - Plurality Winner: The candidate with the most first-place votes.     - Single-Runoff Winner: If no candidate has a majority, the two top vote-getters advance to a second round to determine the winner.     - Pairwise Comparison (Condorcet Method): Every candidate is compared head-to-head against every other candidate. A candidate earns a point for winning a match-up. The winner is the candidate with the most overall points.