Random Variables, Probability Models, Sampling Distributions, and Hypothesis Testing

Random Variables and Probability Models

Definition and Types of Random Variables

A random variable is a numerical value determined by the outcome of a random event. Random variables are fundamental in statistics for modeling uncertainty and variability.

• Discrete Random Variable: Can take on a countable number of distinct values. Example: Number of heads in 10 coin tosses.

• Continuous Random Variable: Can take any value within a given range. Example: Height of students in a class.

Notation: Random variables are typically denoted by capital letters such as X, Y, or Z.

Probability Model

A probability model describes all possible values of a random variable and their associated probabilities.

• For discrete random variables, the model lists each possible value and its probability.

• For continuous random variables, the model specifies a probability density function.

Expected Value and Variance

The expected value (mean) and variance are key properties of random variables.

• Expected Value (Mean):

$\mu = \sum x \cdot P(x)$

• Variance:

$\sigma^2 = \sum (x - \mu)^2 \cdot P(x)$

Bernoulli Trials

A Bernoulli trial is an experiment with only two possible outcomes: success or failure.

• Probability of success: p

• Probability of failure: q = 1 - p

• Each trial is independent.

• Examples: Tossing a coin, yes/no survey responses, basketball free throws.

Uniform Model

If a random variable X can take values 1, 2, ..., n, and each outcome is equally likely, X has a discrete uniform distribution U[1,...,n].

$P(X = i) = \frac{1}{n}$

Standardization (Z-score)

The Z-score measures how many standard deviations a value is from the mean.

$z = \frac{x - \mu}{\sigma}$

Example: To find $P(X > 24)$, calculate the Z-score and use the normal distribution table.

$P(Z > 2.0) = 1 - P(Z \leq 2.0) = 1 - 0.9772 = 0.0228$

Sum of Random Variables

When adding independent random variables, their expected values and variances add:

• Expected Value:

$E(T) = E(P_1) + E(P_2)$

Example: If $E(P_1) = 9$ and $E(P_2) = 9$, then $E(T) = 18$.

• Variance:

$Var(P_1 + P_2) = Var(P_1) + Var(P_2)$

Example: If $\sigma = 1.5$ for both, then $Var(P_1 + P_2) = 1.5^2 + 1.5^2 = 4.5$.

Distribution of Sample Proportions

Assumptions and Conditions

To use the sampling distribution of sample proportions, certain conditions must be met:

• Independence Assumption: Sampled values must be independent.

• Sample Size Assumption: Sample size n must be large enough.

• Randomization Condition: Data should come from a randomized experiment or a simple random sample.

• 10% Condition: If sampling without replacement, n should be no more than 10% of the population.

• Success/Failure Condition: Both $np$ and $nq$ should be at least 10.

The Central Limit Theorem (CLT)

Statement and Implications

The Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population's shape.

• For highly skewed distributions, larger sample sizes (dozens or hundreds) may be needed for normality.

Application Example

• If only 20 recent graduates' salaries are sampled, concerns include small sample size, unknown population size, and unknown standard deviation.

• Confidence intervals should be based on t-models when the population standard deviation is unknown.

• Increasing sample size (e.g., to 60) makes the confidence interval more precise.

Hypothesis Testing

Null and Alternative Hypotheses

Hypothesis testing begins with a null hypothesis ($H_0$), which assumes no effect or no change. The alternative hypothesis ($H_a$) represents all other possible values.

• Null Hypothesis: $H_0: \text{parameter} = \text{hypothesized value}$

• Alternative Hypothesis: $H_a: \text{parameter} \neq \text{hypothesized value}$

• Example: $H_0: p = 0.2$, $H_a: p \neq 0.2$

Standard Error and Z-score for Proportions

• Standard Error of a Proportion:

$SD(p) = \sqrt{\frac{pq}{n}}$

• Z-score for Proportion:

$Z = \frac{p - p_0}{SD(p)}$

P-value

The p-value is the probability, under the null hypothesis, of obtaining a result at least as extreme as the observed result. It is not the probability that the null hypothesis is true.

• P-value is calculated using the sampling distribution (often normal or t-distribution).

• Excel function: T.DIST.RT can be used for right-tailed t-distribution p-values.

Summary Table: Key Concepts

Concept	Definition	Formula	Example/Application
Random Variable	Numerical outcome of a random event	N/A	Policy payout, coin toss
Expected Value	Mean of random variable	$\mu = \sum x \cdot P(x)$	Average payout
Variance	Spread of random variable	$\sigma^2 = \sum (x - \mu)^2 \cdot P(x)$	Risk assessment
Bernoulli Trial	Experiment with two outcomes	N/A	Coin toss, yes/no survey
Uniform Distribution	All outcomes equally likely	$P(X = i) = \frac{1}{n}$	Dice roll
Z-score	Standardized value	$z = \frac{x - \mu}{\sigma}$	Normal distribution analysis
Central Limit Theorem	Sampling distribution approaches normality	N/A	Mean salary estimation
Hypothesis Testing	Test claim about population	$Z = \frac{p - p_0}{SD(p)}$	Proportion test
P-value	Probability of observed result under $H_0$	N/A	Statistical significance