Podcast
Questions and Answers
What is the mean of the population when throwing a fair six-faced die?
What is the mean of the population when throwing a fair six-faced die?
What is the variance of the population when throwing a fair six-faced die?
What is the variance of the population when throwing a fair six-faced die?
In the context of throwing two dice, what is the standard deviation of the sampling distribution of the mean?
In the context of throwing two dice, what is the standard deviation of the sampling distribution of the mean?
How many different possible values are obtained from the sampling distribution of the mean when throwing two dice?
How many different possible values are obtained from the sampling distribution of the mean when throwing two dice?
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When calculating the mean of the sampling distribution for two dice, which of these calculations is correct?
When calculating the mean of the sampling distribution for two dice, which of these calculations is correct?
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What is the relationship between the mean of the sample mean X̄ and the population mean µX in the context of sampling distributions?
What is the relationship between the mean of the sample mean X̄ and the population mean µX in the context of sampling distributions?
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What happens to the standard error of the mean σX̄ as the sample size n increases?
What happens to the standard error of the mean σX̄ as the sample size n increases?
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Which statement best describes the application of the Central Limit Theorem (CLT) with regards to sample size?
Which statement best describes the application of the Central Limit Theorem (CLT) with regards to sample size?
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In the context of a normal distribution of the amounts of soda in bottles, which of the following would correctly describe the parameters if one bottle is selected?
In the context of a normal distribution of the amounts of soda in bottles, which of the following would correctly describe the parameters if one bottle is selected?
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If you were to sample a population with a known mean and variance, which of the following statements regarding the expected value of the sample mean is correct?
If you were to sample a population with a known mean and variance, which of the following statements regarding the expected value of the sample mean is correct?
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Study Notes
Announcements
- Kahoot! registration is available.
- Mid-term Exam #1 grades will be posted after class.
- Homework #5 is due October 22.
- Homework #6 was posted and is due October 29.
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
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