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STA 291 Spring 2009

This lecture discusses the sampling distribution of the mean and proportion, including the Central Limit Theorem. It also covers converting between probabilities, z-scores, and observations in a normal distribution.

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STA 291 Spring 2009

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  1. STA 291Spring 2009 LECTURE 15 THURSDAY, 26 MARCH

  2. Le Menu • 9 Sampling Distributions 9.1 Sampling Distribution of the Mean 9.2 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. 274, #8.35, 37; p. 275, #49

  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 – Study Tools Chapter 9.1 and 9.2 – OR: Sections 9.1 and 9.2 in the textbook • Suggested problems from the textbook: 9.1 – 9.4, 9.18, 9.30, 9.34

  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 #16 Write your name and section number on your index card. Today’s question:

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