Content To Study

Exam 2 will cover material beginning on Week 5 through Week 8. This includes the following topics:

The Normal Distribution

  1. Know how the mean and standard deviation affect the shape and location of a normal distribution

  2. Know the empirical rule and how it relates to the standard deviation of the normal distribution

  3. Know how to compute and interpret a \(z\)-score and how it relates to the standard normal distribution

\[ z = \frac{x-\mu}{\sigma} \]

Experimental and Sampling Designs

  1. Know the difference between population distribution and sampling distribution

  2. Know how to define and identify the three experimental designs we discussed: Completely randomized design, matched pairs design and randomized block design.

  3. Know the terminology for sampling designs (e.g Sampling Frame, Census, sample survey, etc.

  4. Know how to define and identify the sampling designs we talked about in class: Convenience Sampling, Systematic Sampling, Simple Random Sampling, Cluster Sampling, Stratified Random Sampling

  5. Know the difference between Sampling Error and Margin of Error.

  6. Know the difference between Sampling With Replacement vs Sampling Without Replacement

  7. Know the difference between statistical association and causation

Probability Theory

  1. Be comfortable with the language of probability theory: e.g know how to define Probability, Sample Space, Event, Random Trial

  2. Know the rules we discussed for probability: Complement Rule, Addition Rule, Multiplication Rule

  3. Know how to compute the probability of a Complement of an event, Union of two events, and Intersection of two independent events.

  4. Know how to compute probabilities when sampling with vs without replacement.

  5. Know how to use a Venn Diagram to show the relationships between events

  6. Know what is meant when two random variables are said to be Disjoint (mutually exclusive)

  7. Know what is implied by the Law of Large Numbers (e.g the relative frequency of an event will approach the true probability of an event as the number of observations of the event increases)

  8. Know the definitions of Random variable, and the difference between Discrete Random Variables versus Continuous Random Variables.

  9. Understand what is meant by the Probability distribution of a discrete random variable.

  10. Be able to compute the mean, variance, and standard deviation of a discrete random variable from its probability distribution (when given as a table of values and probabilities).

Rules and equations from of probability theory

The Complement Rule: The probability of the complement of an event is one minus the probability of the event \[ P(A') = 1-P(A) \]

The Addition Rule: The probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection \[P(A\cup B) = P(A)+P(B)-P(A\cap B)\]

The Multiplication Rule For Two Independent Events: The probability of the intersection of two independent events is equal to the product of their individual probabilities \[ P(A\cap B) = P(A)\cdot P(B)\]

The Multiplication Rule For Dependent Events: The probability of the intersection of two dependent events is equal to their conditional probability multiplied by the individual probability of the event being condition on \[ P(A\cap B) = P(A|B)\cdot P(B) \]

Mean of Discrete Random Variable \[ \mu =\sum_x x\cdot P(x) \]

Variance of Discrete Random Variable \[ \sigma^2 =\sum_x (x-\mu)^2\cdot P(x) \]

Standard Devation of Discrete Random Variable \[ \sigma =\sqrt{\sum_x (x-\mu)^2\cdot P(x)} \]

Finding Probabilities, Sampling Distributions, and The CLT

  1. Understand what is meant by a population distribution and a sampling distribution.

  2. Know how to compute probabilities for the Bernoulli, Binomial, and Possion distributions using their probability mass functions (PMF)

  3. Know the mean and standard deviation for the Bernoulli, Binomial, and Poisson random variables

  4. Know how to compute probabilities using the probability distribution of a discrete random variable.

  5. Know that the probability of a continuous variable corresponds to the area under the curve of its probability density function

  6. Know the definition of the cumulative distribution of a random variable (e.g \(P(X\leq z)\))

  7. Know how to compute probabilities from a normal probability distribution using a \(Z\)-table (standard normal probability table).

  8. Know how to derive the sampling distribution of the sample mean \(\bar{x}\)

  9. Know how to derive the sampling distribution of the sample proportion \(\hat{p}\)

  10. Understand the central limit theory and how it relates to the sampling distribution of \(\bar{x}\) and \(\hat{p}\)

Special Discrete Distributions

  • Be sure you are comfortable with finding probabilities from the following probability distributions:

Bernoulli Distribution - It models a random experiment with two possible outcomes, often referred to as success and failure. The distribution is characterized by a single parameter, usually denoted as \(p\), which represents the probability of success.

\[PMF: \ \ P(x = k) = \begin{cases} p & \text{if} \ k = 1 \\ (1-p) & \text{if} \ k = 0 \end{cases} \]

  • \(p\) the probability of success

  • the mean is given by \(E[X] = p\)

  • the variance is given by \(Var[X] = p(1-p)\)

The Binomial Distribution - The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure.

\[PMF: \ \ P(x = k) = \frac{n!}{k!(n-k)!}\cdot p^k \cdot (1-p)^{n-k} \]

  • \(p\) - probability of success

  • \(k\) - number of successes

  • \(n\) - number of independent trials (e.g sample size)

  • \(\frac{n!}{k!(n-k)!}\) - the number of ways to have \(k\) successes arranged among \(n\) trials

  • the mean is given by \(E[X] = np\)

  • the variance is given by \(Var[X] = np(1-p)\)

\[ P(X \leq a) = \sum_{k = 0}^a \frac{n!}{k!(n-k)!}\cdot p^k \cdot (1-p)^{n-k} \]

The Poisson Distribution - a discrete probability distribution that models the number of events that occur within a fixed interval of time or space and with fixed rate \(\lambda\)

\[PMF: \ \ P(X =k) = \frac{\lambda^k\cdot e^{-\lambda}}{k!} \]

  • \(\lambda\) - the average rate at which events occur

  • \(k\) - a non-negative integer representing the number of events that occurred

  • \(n\) - number of independent trials (e.g sample size)

  • \(\frac{n!}{k!(n-k)!}\) - the number of ways to have \(k\) successes arranged among \(n\) trials

  • the mean and variance are given by \(E[X] = Var[X] = \lambda\)

\[ P(X \leq a) = \sum_{k = 0}^a \frac{\lambda^k\cdot e^{-\lambda}}{k!} \]





Formulas:

The following formulas will be provided on the the exam

\[\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, \ \ \bar{x} = \frac{1}{n}\sum_{x} xF(x), \ \ \bar{x} = \sum_{x} xRF(X)\]

\[s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2, \ \ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x}^2}\]

\[ z = \frac{x - \mu}{\sigma} \]

\[\mu = \sum_x x\cdot P(x) \]

\[ \sigma^2 = \sum_x (x-\mu)^2\cdot P(x) \]

\[\sigma = \sqrt{\sum_x (x-\mu)^2\cdot P(x)} \]

\[P(A\cup B) = P(A)+P(B)-P(A\cap B) \]

\[P(A\cap B) = P(A)\cdot P(B) \]

\[ P(X > x) = 1 - P(X\leq x) \]

\[ P(a\leq X \leq b) = P(X\leq b) - P(X\leq a)\]

\[ P(x = k) = \frac{n!}{k!(n-k)!}\cdot p^k \cdot (1-p)^{n-k}\]

\[ P(X =k) = \frac{\lambda^k\cdot e^{-\lambda}}{k!}\]