# Summary the sample proportion p hat is a random

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Summary: The sample proportion, p-hat, is a random variable If the sample size n is sufficiently large and the population proportion p isn’t close to either 0 or 1, then this distribution is approximately normal The mean of the sampling distribution is equal to the population proportion p The standard deviation of the sampling distribution is equal to p(1-p)/n Homework: Day 1: pg 588-9; 9.19-21, 24 Day 2: pg 589-91; 9.25-30

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Chapter 9: Sampling Distributions Section 9.3: Sample Means Knowledge Objectives: Students will: State the central limit theorem . Construction Objectives: Students will be able to: Given the mean and standard deviation of a population, calculate the mean and standard deviation for the sampling distribution of a sample mean. Identify the shape of the sampling distribution of a sample mean drawn from a population that has a Normal distribution. Use the central limit theorem to solve probability problems for the sampling distribution of a sample mean. Vocabulary: Central Limit Theorem – the larger the sample size, the closer the sampling distribution for the sample mean from any underlying distribution approaches a Normal distribution Standard error of the mean – standard deviation of the sampling distribution of x-bar Key Concepts: Conclusions regarding the sampling distribution of X-bar: Shape: normally distributed Center: mean equal to the mean of the population Spread: standard deviation less than the standard deviation of the population Mean and Standard Deviation of the Sampling Distribution of x-bar Suppose that a simple random sample of size n is drawn from a large population (sample less than 5% of population) with mean μ and a standard deviation σ. The sampling distribution of x-bar will have a mean μ, x-bar = μ and standard deviation σ x-bar = σ/√n. The standard deviation of the sampling distribution of x-bar is called the standard error of the mean and is denoted by σ x-bar . The shape of the sampling distribution of x-bar if X is normal If a random variable X is normally distributed, the distribution of the sample mean, x-bar, is normally distributed. Central Limit Theorem Regardless of the shape of the population, the sampling distribution of x-bar becomes approximately normal as the sample size n increases. (Caution: only applies to shape and not to the mean or standard deviation)
Chapter 9: Sampling Distributions Central Limit Theorem Regardless of the shape of the population, the sampling distribution of x- bar becomes approximately normal as the sample size n increases. Caution: only applies to shape and not to the mean or standard deviation X or x-bar Distribution Population Distribution Random Samples Drawn from Population x x x x x x x x x x x x x x x x Central Limit Theorem in Action n =1 n = 2 n = 10 n = 25 From Sullivan: “With that said, so that we err on the side of caution, we will say that the distribution of the sample mean is approximately normal provided that the sample size is greater than or equal to 30, if the distribution of the population is unknown or not normal.”

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Chapter 9: Sampling Distributions Shape, Center and Spread of Population Distribution of the Sample Means Shape Center Spread
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