Study Notes: Random Variables, Sampling Distributions, Confidence Intervals, and Hypothesis Testing

Chapter 4: Random Variables and Probability Distributions

Random Variables

A random variable is a numeric value associated with the outcome of a probability experiment. Random variables are classified as either discrete or continuous based on the type of values they can assume.

• Discrete Random Variable: Takes on countable values, often whole numbers, typically associated with counting (e.g., number of heads in coin tosses).

• Continuous Random Variable: Takes on any value within an interval, typically associated with measurement (e.g., weight, time, volume).

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

Key Properties of Discrete Random Variables

• Mean (Expected Value): The mean of a discrete random variable X is given by: $E(X) = \sum x_i p(x_i)$

• Variance: The variance of a discrete random variable X is: $Var(X) = \sum (x_i - \mu)^2 p(x_i)$

• Probability Distribution: Represented by a table or chart listing all possible values and their probabilities.

Key Properties of Continuous Random Variables

• Values are measured, not counted, and can take any value within an interval.

• Probability distribution is represented by a probability density function (PDF), a smooth curve where the total area under the curve equals 1.

• The probability that the variable falls within a certain interval is the area under the curve over that interval.

Key Discrete Random Variables

• Binomial Random Variable:

  • Fixed number of independent trials (n).

  • Each trial has two possible outcomes: success or failure.

  • Probability of success (p) is constant for each trial.

  • Counts the number of successes in n trials.

  • Probability mass function: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

  • Calculator commands: BINOMPDF (exactly X successes), BINOMCDF (X or fewer successes).

• Poisson Random Variable:

  • Counts the number of events (successes) in a fixed interval of time or space.

  • Events occur independently and at a constant average rate (λ, lambda).

  • Probability mass function: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

  • Calculator commands: POISSONPDF (exactly X successes), POISSONCDF (X or fewer successes).

• Hypergeometric Random Variable:

  • Used when sampling without replacement from a finite population with known composition.

  • Example: Drawing marbles of different colors from a jar without replacement.

Normal Random Variable

• The normal distribution is the most important continuous random variable, with a bell-shaped probability density function centered at the mean (μ) and standard deviation (σ).

• Probabilities are computed using the NormalCDF command; percentiles are found using INVNORM.

• The standard normal random variable (Z) has mean 0 and standard deviation 1. Z-scores indicate the number of standard deviations from the mean.

• Useful for comparing values from different normal distributions (e.g., SAT vs. ACT scores).

Example

• Suppose X is the number of heads in 10 coin tosses (binomial, n=10, p=0.5). The probability of exactly 6 heads is: $P(X=6) = \binom{10}{6} (0.5)^6 (0.5)^4$

Chapter 5: Sampling Distributions

Sampling Distributions

A sampling distribution is the probability distribution of a statistic (such as the sample mean or sample proportion) computed from a random sample. It describes how the statistic varies from sample to sample.

• Sample averages (\bar{x}) and sample proportions (\hat{p}) are continuous random variables.

• Their probability distributions are called sampling distributions.

Key Properties

• For the sample mean:

  • Expected value: $E(\bar{x}) = \mu$

  • Standard deviation (standard error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (approximated by $\frac{s}{\sqrt{n}}$ if population σ is unknown)

• For the sample proportion:

  • Expected value: $E(\hat{p}) = p$

  • Standard deviation: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Central Limit Theorem (CLT): For large sample sizes (n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• If the population is normal, the sampling distribution of the sample mean is normal for any sample size.

• Larger samples yield tighter (less variable) sampling distributions; standard error decreases as n increases.

• Unbiased Estimator: An estimator whose expected value equals the population parameter (e.g., sample mean for population mean).

• Minimum Variance: Among unbiased estimators, the one with the smallest variance is preferred.

Example

• If the population mean is 100 and standard deviation is 15, for samples of size 36: $\sigma_{\bar{x}} = \frac{15}{\sqrt{36}} = 2.5$

Chapter 6: Confidence Intervals for μ and p

Confidence Intervals

A confidence interval (CI) is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence (e.g., 95%).

• Confidence Level (CL): The probability that the CI contains the parameter in repeated samples (e.g., 0.95, 0.90, 0.99).

• Margin of Error: The half-width of the confidence interval; reflects sampling variability.

Large Sample Confidence Interval for μ

• Relies on the Central Limit Theorem; use when n is large (n ≥ 30).

• Formula: $\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$

• zα/2 is found using the invnorm calculator command, with α = 1 – CL.

Small Sample Confidence Interval for μ

• Use when n is small and the population is approximately normal.

• Formula: $\bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}}$

• tα/2, df is found using the invt calculator command, with degrees of freedom (df) = n – 1.

Confidence Interval for p (Proportion)

• Large sample: At least 15 successes and 15 failures in the sample.

• Formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Calculator command: 1prop-ZINT.

• For small samples, special methods are required (no calculator command).

Sample Size Determination

• To achieve a desired margin of error, solve for n in the margin of error formula.

• If p is unknown, use p = 0.5 for maximum variability.

Alpha (α)

• α = 1 – confidence level; determines the critical z or t values for the CI.

• The area in each tail is α/2.

Example

• For a sample mean of 50, s = 10, n = 100, and 95% confidence: $50 \pm 1.96 \times \frac{10}{10} = 50 \pm 1.96$ CI: (48.04, 51.96)

Chapter 7: Hypothesis Testing for μ and p

Hypothesis Testing

A hypothesis test is a statistical procedure to test claims about population parameters using sample data.

Key Components

• Null Hypothesis (H0): The status quo or default claim; always contains an equality (e.g., μ = μ0).

• Alternative Hypothesis (Ha): The claim we seek evidence for; uses >, <, or ≠.

• Type I Error (α): Rejecting H0 when it is true.

• Type II Error (β): Failing to reject H0 when it is false.

• Significance Level (α): Probability of a Type I error; chosen by the researcher (e.g., 0.05).

• Test Statistic: The sample statistic converted to a z or t value.

• Critical Value: The z or t value that marks the boundary of the rejection region.

• Rejection Region: The set of values for which H0 is rejected.

• P-value: The probability, under H0, of observing a result as extreme as the sample result.

Types of Tests

• One-tailed Test (Left): Ha: parameter < value

• One-tailed Test (Right): Ha: parameter > value

• Two-tailed Test: Ha: parameter ≠ value

Decision Rules

• If p-value < α, reject H0; sufficient evidence for Ha.

• If p-value > α, fail to reject H0; insufficient evidence for Ha.

• If test statistic falls in the rejection region, reject H0.

Reporting Decisions

• "There is sufficient evidence at α = xx to reject the null hypothesis and accept the alternative hypothesis."

• "There is insufficient evidence at α = xx to reject the null hypothesis. Therefore, the null hypothesis is plausible."

Test Types and Calculator Commands

• Large Sample Test for μ: Use normal distribution (ZTEST).

• Small Sample Test for μ: Use t distribution (TTEST), if population is normal.

• Large Sample Test for p: Use normal distribution (1propZtest), requires at least 15 successes and 15 failures (n*p0 ≥ 15, n*(1–p0) ≥ 15).

Example

• Suppose H0: μ = 100, Ha: μ > 100, sample mean = 105, s = 10, n = 25, α = 0.05. Test statistic: $t = \frac{105 - 100}{10/\sqrt{25}} = 2.5$ Compare t to critical value from t-table with df = 24.

Summary Table: Key Random Variables

Random Variable	Context	Parameters	Mean	Variance	Calculator Command
Binomial	Fixed number of trials, success/failure	n, p	$np$	$np(1-p)$	BINOMPDF, BINOMCDF
Poisson	Events in interval (time/space)	λ	$\lambda$	$\lambda$	POISSONPDF, POISSONCDF
Hypergeometric	Sampling without replacement	N, K, n	$n \frac{K}{N}$	$n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1}$	--
Normal	Measurement, continuous	μ, σ	$\mu$	$\sigma^2$	NormalCDF, INVNORM

Additional info: Table entries for mean and variance of the hypergeometric distribution are standard formulas inferred for completeness.