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Sampling Distribution

• Lecture 7
• Topic: Sampling Distribution
• Instructor: Shouqiang Wang
• Course: OPRE 6301: Statistics and Data Analysis
• Location: University of Texas at Dallas

Statistics and Data Analysis

• Topic: Descriptive Statistics, Inferential Statistics, and Predictive Statistics
• Concepts: includes graphical and numerical exploratory techniques, estimation, hypothesis testing, inference on one, two, and more populations (ANOVA).
• Sub-topics: Sampling Distributions, Continuous Distributions, Discrete Distributions, Probability & Random Variable, Simple Linear Regression, Multiple Linear Regression
• Methodologies: Data collection and sampling, sampling distributions

Outline

• Topic: Sampling Distribution of the Mean
• Topic: Sampling Distribution of a Proportion

Example: Throwing a Die

• Variable: X - the result of throwing a fair six-sided die.
• Probability Distribution: Each outcome has a probability of 1/6.
• Population Mean (μx): 3.5
• Population Variance (σ²x): 2.92
• Population Standard Deviation (σx): 1.71

Sampling Distribution of Throwing Two Dice

• Sample Occurrences: Each sample combination has a probability of 1/36.
• Sample Mean (X): Values range from 1.0 to 6.0.
• 11 Possible Values of X: 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0

Sampling Distribution of the Mean of Two Dice

• Mean (μx): 3.5
• Variance (σ²x): 1.46
• Standard Deviation (σx): 1.21

Distribution of X

• μx = μx
• σx = σx / √2

Properties of Sample Mean Distribution

• Centered at population mean (μx): The distribution's center is at the population mean.
• Tightens with increasing sample size (n): The distribution becomes more concentrated around the mean as the sample size increases.
• Approaches bell-shaped curve with larger n: The shape of the distribution approaches a bell-shaped normal distribution as the sample size (n) increases.

Central Limit Theorem

• Sample means that are normally distributed regardless of population distribution whenever the sample size is large enough.
• Expected value of the sample mean equals the population mean (no bias).
• Standard error equals the population std deviation / √n (precision increases with sample size (n)).

Probability of a Bottle Containing More Than 32 Ounces

• Distribution: Normal distribution with mean (μ) = 32.2 ounces and standard deviation (σ) = 0.3 ounce.
• Probability: 0.7486

Probability of Mean Amount of Four Bottles Exceeding 32 Ounces

• Distribution: Normal distribution with mean (μx) = 32.2 ounces and standard deviation (σx) = .3/√4 = 0.15 ounces.
• Probability: 0.9082

Sampling Distribution of Difference Between Two Means

• Mean: μ1 - μ2
• Standard deviation: √(σ²1/n1 + σ²2/n2)

Mean Starting Salaries of UTD and SMU Graduates

• UTD Graduates: mean ($70,000), std deviation ($15,000)
• SMU Graduates: mean ($68,000), std deviation ($16,000)
• Sample sizes: 60 (UTD), 50 (SMU)
• Probability: 0.7490

Binomial Probabilities

• Definition: A probability distribution of the number of successes in a fixed number of independent trials with a constant probability of success.
• Formula: P(k successes in n trials) = (n! / (k! * (n − k)!)) * p^k * (1 − p)^(n − k)

Facebook Users

• Power Users: A subset of Facebook users significantly more active than normal, contributing more content & interaction.

Normal Approximation to Binomial

• Formula: Bin(n, p) ≈ N(μ = np, σ = √np(1 − p))
• Conditions: Use when the number of trials (n) is large enough

Correction Factor for Normal Approximation

• Correction is sometimes applied.
• Correction is usually omitted when n is large.

Determining Large Sample Sizes: Binomial

• Rule: np ≥ 5 and n(1 − p) ≥ 5 is a common rule to use in applying binomial approximation; a larger threshold like 10 might be used in some cases.

Sampling Distribution of a Sample Proportion

• Distribution: Approximately normally distributed.
• Mean: μp = p
• Standard Error: σp = √p(1 − p)/n

Probability for Defective Phone Rate

• Distribution: The daily number of defective phone products and sample proportion of defectives follow a normal distribution.
• Result: 0.152