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**Unformatted text preview: **Example 5: We can check for undercoverage or nonresponse by comparing the sample proportion to the population proportion. About 11% of American adults are black. The sample proportion in a national sample was 9.2%. Were blacks underrepresented in the survey? 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 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...

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