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Understanding Sampling Distributions and the Central Limit Theorem in Statistical Methods

In this lecture, we explore the concept of sampling distributions through a practical example involving a cereal manufacturer's weight settings for their products. We address the normal distribution of cereal weights and calculate the percentage of boxes underweight. Furthermore, the session covers the properties of sampling distributions, including expected values and standard deviations. The Central Limit Theorem is introduced, illustrating how larger sample sizes lead to a normal distribution pattern, impacting decision-making and statistical analysis.

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Understanding Sampling Distributions and the Central Limit Theorem in Statistical Methods

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  1. STA291 Statistical Methods Lecture 15

  2. Lecture 14 Review • A cereal manufacturer labels their cereal boxes “16o oz.” To prevent underweight boxes, the manufacturer sets the packaging machine to 16.3 oz. The company believes that the amount of cereal in the boxes fits a normal distribution, N(16.3, 0.2^2). • A) What percentage of boxes will weigh less than 16 oz.? • B) If the company lawyer insists that no more than 4% of boxes can be underweight, what mean setting should the company use?

  3. 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

  4. 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…

  5. Sampling Distribution: Part Two • 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

  6. 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…

  7. Looking back • Sampling distribution (of any statistic): • center • spread • shape • Central Limit Theorem (CLT)

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