Week 1, Day 3, Part 1-Scientific Notation, Inflation, CPI, and Contingency Tables

Logistics and Course Progress

Overview of Recent Topics: The past two days of class have covered percent, decimal, and fraction form changes, absolute and relative change, and the concept of scale with regards to the power of ten (very large and very small values).
Quiz Details: A quiz will be held tomorrow toward the end of class.
- Duration: Specifically designed to take $25$ to $30$ minutes, though students can take longer if needed.
- Topics: The four aforementioned topics (percent/decimal/fraction changes, absolute change, relative change, and scale/power).
- Announcement: A direct announcement will be posted on Canvas specifying the number of questions for each topic.
Exam 1: Material introduced today (Thursday) will not be on tomorrow's quiz but will be included in Exam 1, scheduled for next Thursday.

Scientific Notation Review

Mathematical Form: Scientific notation always takes the form of $a \times 10^n$ , where $a$ is a number such that $1 \le a < 10$ .
Warm-up Problems:
- Standard Form to Scientific Notation: For the number $00071$ :
- The leading zeros are ignored because $71$ is the same number regardless of how many zeros are placed in front of it.
- The decimal point starts at the end ( $71.$ ) and moves one place to the left to obtain $7.1$ .
- Result: $7.1 \times 10^1$ .
- Scientific Notation to Standard Form: For the number $-3.7 \times 10^4$ :
- To convert to standard form, the decimal indicates movement four places to the right.
- Moving the decimal from behind the $3$ results in: $37,000$ .
- The negative sign from the coefficient is tacked onto the final answer.
- Result: $-37,000$ .
- Operations (Multiplication): To multiply $(3.2 \times 10^2) \times (2.0 \times 10^4)$ :
- Group coefficients and exponents separately: $(3.2 \times 2.0) \times (10^{2+4})$ .
- Multiply coefficients: $3.2 \times 2.0 = 6.4$ .
- Add exponents for base $10$ numbers: $2 + 4 = 6$ .
- Result: $6.4 \times 10^6$ .
Q&A Regarding Exponents:
- If the exponent is negative (e.g., $10^{-4}$ ), the decimal moves in the opposite direction (left), creating a number less than one (e.g., $0.00037$ ).
- Positive exponents indicate a number greater than one in standard form.
- A negative sign in front of the coefficient only indicates that the final number is negative; it does not change the process of moving the decimal.

Absolute and Relative Change Review

Formulas:
- Absolute Change: $A - B$ (Result - Reference).
- Relative Change: $\frac{A - B}{B}$ .
Detroit Case Study (Population Change):
- 1968 Population ( $B$ ): $6,750,000$ .
- 2006 Population ( $A$ ): $3,375,000$ .
- Calculation:
- Absolute Change: $3,375,000 - 6,750,000 = -3,375,000$ .
- Relative Change: $\frac{-3,375,000}{6,750,000} = -0.5$ , which is equal to $-50\%$ .
Contextual Interpretation: The negative result demonstrates a decrease in population over $38$ years. If a question specifically asks "how much did the population decrease," the negative sign might be omitted in the verbal answer (e.g., "it decreased by $50\%$ "), but the mathematical change is negative.

Index Numbers and Consumer Price Index (CPI)

Index Numbers: A direct statistical measurement used to track the change in a variable (typically the price of goods) over time.
- Higher numbers in later years indicate a percentage increase.
- Used for various economic variables.
Consumer Price Index (CPI):
- A specific index based on the average monthly prices of approximately $60,000$ items people buy in daily life (e.g., shampoo, conditioner, cat food, airplane tickets, laundry soap, blankets, home repairs, clothing, computer parts).
- It provides a holistic snapshot of the cost of living and inflation.
Index Number Calculation: $\text{Index Number} = \left( \frac{\text{Current Price}}{\text{Reference Year Price}} \right) \times 100$ .

Inflation

Definition: The general level of price increases over time. As the CPI number goes up, inflation goes up.
Inflation as Investment: Inflation can be beneficial for those who own assets like housing. Houses are often treated as investments where value increases over $15$ to $20$ years, allowing owners to sell for profit.
Purchasing Power: If wages do not increase at the same rate as inflation, purchasing power decreases.
- Example: If a salary increase is only $1\%$ but inflation is $3\%$ , the individual is effectively losing money relative to the cost of goods.
Causes of Inflation:
- Supply and Demand: Scarcity of goods drives prices up (e.g., toilet paper and hand sanitizer during the COVID-19 pandemic).
- Cost-Push Inflation: Occurs when the cost of production rises (e.g., oil imports from Iran becoming difficult to obtain, leading to higher gas and transportation prices).
- Built-In Inflation: Predictive inflation based on economic models or future expectations.
- Example (Boeing): If news of manufacturer issues or falling airplane parts threatens a company's future profits, airlines may limit flights and cut workers in anticipation, causing economic shifts before the event fully impacts the public.

Calculating Inflation with CPI

Methodology: The inflation rate is measured using the Relative Change formula between two different years.
Example Comparison (1988–1989):
- CPI 1989 ( $A$ ): $124.0$ .
- CPI 1988 ( $B$ ): $118.3$ .
- Calculation: $\frac{124.0 - 118.3}{118.3} \approx 4.8\%$ .
- Non-Eyeball Rule: You cannot simply subtract the index numbers (e.g., $124 - 118.3 = 5.7$ ) and call it the inflation rate because the reference point (denominator) is not $100$ .

Real and Nominal Values

Nominal Value: Current prices listed without considering the impact of inflation.
Real Value: Adjusts the nominal value to account for inflation, representing the actual worth or purchasing power.
- Example: A $5\%$ pay increase in a year with $2\%$ inflation results in a "real" income increase of only $3\%$
- Negotiation Tip: Historically, inflation averages about $3\%$ per year; therefore, a salary increase should be at least $3\%$ to maintain the same standard of living.

CPI Formula for Price Conversion

Goal: To find out how much a specific dollar amount from one year is worth in another year.
Formula: $\text{New Price} = \text{Old Price} \times \left( \frac{\text{New CPI}}{\text{Old CPI}} \right)$ .
Example (1995 to 2005):
- CPI 1995: $152$ .
- CPI 2005: $195$ .
- Old Price (1995): $\$1,000$ .
- Calculation: \1,000 \times \left( \frac{195}{152} \right) = \ $1,282.89$ .

Contingency Tables and Probability

Definition: Tables that display cases and combinations of two discrete variables (e.g., Biological Sex vs. Animal Preference).
Structure:
- Cells: Intersections of variables (e.g., Women who prefer dogs).
- Margins: Totals found in the rightmost column and bottom row.
- Overall Total ( $N$ ): The number in the bottom-right corner representing the entire survey group.
Types of Probability:
- Marginal Probability: Summarizes one variable independently of the other. The numerator is found in the margins, and the denominator is the overall total ( $N$ ).
- Example (Passing Rates): If $60$ out of $100$ students passed, the marginal probability of passing is $0.60$ .
- Joint Probability: Uses the word "and." The numerator is an intersection cell (center of the table), and the denominator is the overall total ( $N$ ).
- Example: Probability someone studied at least $6$ hours and passed: $\frac{36}{100} = 0.36$ .
- Conditional Probability: Marked by the word "given." This is the only type that does not use the overall total as the denominator.
- Formula: The denominator is the total of the condition specified after the word "given."
- Example: Probability a student passes given they studied at least $6$ hours.
  - Total who studied $6$ hours: $45$ .
  - Number of those who passed: $36$ .
  - Result: $\frac{36}{45} = 0.80$ (or $80\%$ ).
Expansion: Tables can be larger (e.g., Age ranges vs. Movie genres like Horror, Comedy, Romance), but the logic for the three probability types remains constant.