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STA 291 Fall 2009

This lecture delves into the fundamental concepts of sampling distributions, specifically focusing on the sampling distribution of the mean and the proportion. Key topics include the Central Limit Theorem (CLT), illustrating how the distribution of sample means approaches normality as the sample size increases. We cover reverse processes in probability calculations, typical distribution questions, and important properties like expected value and standard error. Engage with example exercises to solidify your understanding and gain insights into statistical methodologies used for analyzing sample data.

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STA 291 Fall 2009

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  1. STA 291Fall 2009 LECTURE 22 TUESDAY, 10 November

  2. Le Menu • 9 Sampling Distributions 9.1 Sampling Distribution of the Mean 9.5 Sampling Distribution of the Proportion Including the Central Limit Theorem (CLT), the most important result in statistics • Homework Saturday at 11 p.m.

  3. Going in Reverse, S’More • What about “non-standard” normal probabilities? Forward process: x  z  prob Reverse process: prob  z  x • Example exercises: p. 249, #1 to #12

  4. Typical Normal Distribution Questions • One of the following three is given, and you are supposed to calculate one of the remaining 1. Probability (right-hand, left-hand, two-sided, middle) 2. z-score 3. Observation • In converting between 1 and 2, you need Table 3. • In transforming between 2 and 3, you need the mean and standard deviation

  5. Chapter 9 Points to Ponder • Suggested Reading Sections 9.1 to 9.5 in the textbook • Suggested problems from the textbook: 9.1 – 9.14,

  6. Chapter 9: Sampling Distributions • If you repeatedly take random samples and calculate the sample mean each time, the distribution of the sample mean follows a pattern • This pattern is the sampling distribution Population with mean m and standard deviation s

  7. Properties of the Sampling Distribution • Expected Value of the ’s: m. • Standard deviation of the ’s: also called the standard error of • (Biggie) Central Limit Theorem: As the sample size increases, the distribution of the ’s gets closer and closer to the normal. Consequences…

  8. Example of Sampling Distribution of the Mean As n increases, the variability decreases and the normality (bell-shapedness) increases.

  9. Sampling Distribution: Part Deux • If you repeatedly take random samples and calculate the sample proportion each time, the distribution of the sample proportion follows a pattern Binomial Population with proportion p of successes

  10. Properties of the Sampling Distribution • Expected Value of the ’s: p. • Standard deviation of the ’s: also called the standard error of • (Biggie) Central Limit Theorem: As the sample size increases, the distribution of the ’s gets closer and closer to the normal. Consequences…

  11. Example of Sampling Distributionof the Sample Proportion As n increases, the variability decreases and the normality (bell-shapedness) increases.

  12. Central Limit Theorem • Thanks to the CLT … • We know is approximately standard normal (for sufficiently large n, even if the original distribution is discrete, or skewed). • Ditto

  13. Attendance Question #22 Write your name and section number on your index card. Today’s question:

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