Probability and the Standard Normal Distribution Notes

The syllabus allocates a significant amount of time to Chapter Six, specifically during the Fall 2020 semester.
The schedule includes:
- Two standard lectures.
- Two laboratory sessions.
- At least one additional "extra credit" day.
This extensive scheduling is due to the complexity of the problems introduced regarding probability and distributions.

Discrete Variables:
- These variables consist of separate, indivisible categories.
- Decisions or measurements are "this or that," with no values possible in between.
- Example 1: Academic Majors. An individual is a Psychology, Sociology, Business, or Nursing major. They cannot be "Sociology 0.5."
- Example 2: Number of Siblings. One can have $0$ , $1$ , or $2$ siblings, but cannot have $2.3$ siblings.
- Visual Representation: Technically, discrete variables should be represented by a bar graph with spaces between the bars to indicate the distinct nature of the categories, rather than a histogram where bars touch.
Continuous Variables:
- These variables can be divided into an infinite number of fractional parts; the decimal places can theoretically go on forever.
- Example 1: Weight and Height. While someone might say they weigh $180\text{ lbs}$ , a more precise measurement might be $180.1\text{ lbs}$ or $180.13\text{ lbs}$ .
- Example 2: Hair Length. Hair does not jump from $1\text{ inch}$ to $2\text{ inches}$ overnight; it must grow through every intermediate value (e.g., $1.1$ , $1.2$ , $1.3$ ).
- Visual Representation: These are typically represented by a smooth curve (relative frequency) because the values run together.

In a discrete distribution, probability is determined by counting the frequency of specific outcomes relative to the total number of observations ( $N$ ).
Case Study: Quiz Scores.
- A distribution of $10$ quizzes ( $N = 10$ ) with possible scores ( $X$ ) from $1$ to $6$ :
- Score of $1$ : $2$ people.
- Score of $2$ : $1$ person.
- Score of $3$ : $2$ people.
- Score of $4$ : $3$ people.
- Score of $5$ : $1$ person.
- Score of $6$ : $1$ person.
- Probability Calculations:
- Finding $P(X > 4)$ : Only scores of $5$ and $6$ fit. There are $2$ such scores out of $10$ .
  - $P = \frac{2}{10} = 0.2 = 20\%$
- Finding $P(X ≥ 4)$ : This includes scores of $4$ , $5$ , and $6$ . There are $5$ such people.
  - $P = \frac{5}{10} = 0.5 = 50\%$
- Finding $P(X < 3)$ : This includes scores of $1$ and $2$ . There are $3$ such people.
  - $P = \frac{3}{10} = 0.3 = 30\%$
- Finding $P(X ≥ 1)$ : Every score in this specific distribution is $1$ or higher.
  - $P = \frac{10}{10} = 1.0 = 100\%$

For continuous variables, you cannot simply count individual blocks because values are infinite and fractional.
The Rule of Area: Probability in a continuous distribution is found by measuring the area under the curve between specific points.
Mathematical Context:
- Measuring the area under a curve between two points is known as Integration in Calculus.
- However, Calculus is not a prerequisite for this course. Students will use a pre-calculated table to find these areas.
Visualizing Probability:
- If you mark a point (e.g., $X = 4$ ) and want the probability of a score greater than that, you shade the entire region to the right of that point.
- The probability is the proportion of the shaded area relative to the total area under the curve.
- If the shaded area constitutes $42\%$ of the total area, then $P(X > 4) = 0.42$ or $42\%$ .
- Probability and area are essentially interchangeable concepts in this context.

This is a specific type of distribution used for the majority of the problems in this chapter.
Definition of "Standard": It uses $z$ -scores ( $z$ ) rather than raw scores ( $X$ ).
- Raw scores (dollars, pounds, inches) are converted using the formula: $z = \frac{X - μ}{σ}$ .
Definition of "Normal": The distribution possesses a symmetrical, bell-shaped curve.
Key Characteristics of the Standard Normal Distribution:
- The mean ( $μ$ ) always has a $z$ -score of $0$ .
- The standard deviation ( $σ$ ) is $1$ .
- It is perfectly symmetrical: $50\%$ ( $0.50$ ) of the area lies above the mean, and $50\%$ ( $0.50$ ) lies below the mean.
- $z$ -scores represent the number of standard deviations a score is from the mean.
- $z = 1.0$ is one standard deviation above the mean.
- $z = -2.0$ is two standard deviations below the mean.
- $z = 3.0$ is considered an extreme outlier.

The Unit Normal Table (often called the " $z$ -table") allows students to find probabilities without performing calculus.
Location Resources:
- Course Packet: Problems on page 30; Statistical tables on page 222.
- Textbook ( $8^{th}$ Edition): Table located around page 571.
Two Primary Types of Problems:
1. Given a $z$ -score, find the proportion ( $p$ ): Looking up a specific $z$ value to determine the shaded area (probability) above or below it.
2. Given a proportion ( $p$ ), find the $z$ -score: Finding a specific percentage (e.g., "the top $20\%$ ") and determining which $z$ -score cuts off that area.

$z$ -Scores:
- Can be positive, negative, or zero.
- Represent distance from the mean in standard deviation units.
- $z = 0$ : Exactly at the mean.
- $z = 1.2$ : $1.2$ standard deviations above the mean.
- $z = -2.3$ : $2.3$ standard deviations below the mean.
Proportions ( $p$ ) / Probabilities:
- Referred to interchangeably as proportion, percent, probability, or area.
- Range: Must be between $0$ and $1$ (as a proportion) or $0\%$ and $100\%$ (as a percent).
- Negativity Rule: Probability or area cannot be negative. A negative probability is impossible.
- Maximum Value: The highest possible proportion is $1.0$ ( $100\%$ ), representing a "sure thing" or the total area under the curve.