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Sampling Distributions Prof. Jacob M. Montgomery Quantitative Political Methodology (L32 363) September 14, 2016 Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 1/6 Sampling Distributions Sampling Distributions A sampling distribution is the distribution of a statistic given repeated sampling. Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 2/6 Sampling Distributions Sampling Distributions A sampling distribution is the distribution of a statistic given repeated sampling. We use probability theory to derive a distribution for a statistic Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 2/6 Sampling Distributions Sampling Distributions A sampling distribution is the distribution of a statistic given repeated sampling. We use probability theory to derive a distribution for a statistic, which allows us (eventually) to make statistical inferences about population parameters. Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 2/6 Central limit theorem Central limit theorem For random sampling with a large sample size n, the samplingdistribution of the sample mean ȳ is approximately normal, where ȳ ∼ N µ, √σn . Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 3/6 Central limit theorem Central limit theorem For random sampling with a large sample size n, the samplingdistribution of the sample mean ȳ is approximately normal, where ȳ ∼ N µ, √σn . Some notes: As n → ∞, the standard error is going to get smaller and smaller. Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 3/6 Central limit theorem Central limit theorem For random sampling with a large sample size n, the samplingdistribution of the sample mean ȳ is approximately normal, where ȳ ∼ N µ, √σn . Some notes: As n → ∞, the standard error is going to get smaller and smaller. This is why the normal distribution is so very important. Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 3/6 Central limit theorem Central limit theorem For random sampling with a large sample size n, the samplingdistribution of the sample mean ȳ is approximately normal, where ȳ ∼ N µ, √σn . Some notes: As n → ∞, the standard error is going to get smaller and smaller. This is why the normal distribution is so very important. Usually n=30 is “good enough”, but it will depend on the distribution. Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 3/6 It works for EVERYTHING Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 4/6 Sampling Distribution of ȳ 1 A key common statistic is ȳ = ∑ yi for a single sample, where n is the n sample size. How is this statistic distributed? Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 5/6 Sampling Distribution of ȳ 1 A key common statistic is ȳ = ∑ yi for a single sample, where n is the n sample size. How is this statistic distributed? The mean of the distribution is known to be µ (the population mean). Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 5/6 Sampling Distribution of ȳ 1 A key common statistic is ȳ = ∑ yi for a single sample, where n is the n sample size. How is this statistic distributed? The mean of the distribution is known to be µ (the population mean). What about the spread? Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 5/6 Sampling Distribution of ȳ 1 A key common statistic is ȳ = ∑ yi for a single sample, where n is the n sample size. How is this statistic distributed? The mean of the distribution is known to be µ (the population mean). What about the spread? Standard error The standard deviation of the sampling distribution of ȳ , denoted σȳ , is σ called the standard error of ȳ , and is equal to √ . n Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 5/6 Sampling Distribution of ȳ 1 A key common statistic is ȳ = ∑ yi for a single sample, where n is the n sample size. How is this statistic distributed? The mean of the distribution is known to be µ (the population mean). What about the spread? Standard error The standard deviation of the sampling distribution of ȳ , denoted σȳ , is σ called the standard error of ȳ , and is equal to √ . n Under certain circumstances we can safely assume that ȳ ∼ N (µ, √σn ). Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 5/6 Class activity 1 On a sheet of paper, write all of the names for your group. 2 Take a sample of 25 from your bag. 3 Calculate the mean of the sampling distribution. Add it to the dot plot on the board. 4 Estimate the population mean. 5 Estimate the population variance. 6 Estimate the sampling distribution of the mean. 7 In a sample you drew, what does each data point represent? In a sampling distribution on the board, what does each data point represent? Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 6/6 Class business Attendance Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 7/6 Class business Attendance Next class we are going to learn how to calculate probabilities for a known distribution Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 7/6 Class business Attendance Next class we are going to learn how to calculate probabilities for a known distribution Then we pull it all together to make our first true inference Lecture 5 (QPM 2016) Sampling Distributions September 14, 2016 7/6