1 / 13

Chapter 11

Chapter 11. Sampling Distributions. Parameters and Statistics. Parameter ≡ a constant that describes a population or probability model , e.g., μ from a Normal distribution Statistic ≡ a random variable calculated from a sample e.g., “x-bar” These are related but are not the same!

thom
Download Presentation

Chapter 11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 11 Sampling Distributions Sampling Distributions

  2. Parameters and Statistics • Parameter ≡ a constant that describes a populationor probability model, e.g., μ from a Normal distribution • Statistic ≡ a random variable calculated from a sample e.g., “x-bar” • These are related but are not the same! • For example, the average age of the SJSU student population µ = 23.5 (parameter), but the average age in any sample x-bar (statistic) is going to differ from µ Sampling Distributions

  3. Example: “Does This Wine Smell Bad?” • Dimethyl sulfide (DMS) is present in wine causing off-odors • Let X represent the threshold at which a person can smell DMS • X varies according to a Normal distribution with μ= 25 and σ= 7 (µg/L) Sampling Distributions

  4. Law of Large Numbers This figure shows results from an experiment that demonstrates the law of large numbers (will be discussed in class) Sampling Distributions

  5. Sampling Distributions of Statistics • The sampling distribution of a statistic predicts the behavior of the statistic in the long run • The next slide show a simulated sampling distribution of mean from a population that has X~N(25, 7). We take 1,000 samples, each of n =10, from population, calculate x-bar in each sample and plot. Sampling Distributions

  6. Simulation of a Sampling Distribution of xbar Sampling Distributions

  7. x-bar is an unbiased estimator of μ Square root law μand σof x-bar Sampling Distributions

  8. Sampling Distribution of Mean Wine tasting example Population X~N(25,7) Sample n = 10 By sq. root law, σxbar = 7 / √10 = 2.21 By unbiased property, center of distribution = μThusx-bar~N(25, 2.21) Sampling Distributions

  9. Illustration of Sampling Distribution: Does this wine taste bad? What proportion of samples based on n = 10 will have a mean less than 20? • State: Pr(x-bar ≤ 20) = ?Recall x-bar~N(25, 2.21) when n = 10 • Standardize: z = (20 – 25) / 2.21 = -2.26 • Sketch and shade • Table A: Pr(Z < –2.26) = .0119 Sampling Distributions

  10. Central Limit Theorem No matter the shape of the population, the distribution of x-bars tends toward Normality Sampling Distributions

  11. Central Limit Theorem Time to Complete Activity Example Let X ≡ time to perform an activity. Xhas µ = 1 & σ = 1 but is NOT Normal: Sampling Distributions

  12. Central Limit Theorem Time to Complete Activity Example These figures illustrate the sampling distributions of x-bars based on • n = 1 • n = 10 • n = 20 • n = 70 Sampling Distributions

  13. xbar 0 z -1.42 Central Limit Theorem Time to Complete Activity Example The variable X is NOT Normal, but the sampling distribution of x-bar based on n = 70 is Normal with μx-bar = 1 and σx-bar = 1 / sqrt(70) = 0.12, i.e., xbar~N(1,0.12) What proportion of x-bars will be less than 0.83 hours? (A) State: Pr(xbar < 0.83) (B) Standardize: z = (0.83 – 1) / 0.12 = −1.42 (C) Sketch: right (D) Pr(Z <−1.42) = 0.0778 Sampling Distributions

More Related