Sampling Distributions
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1 Sampling Distributions Prof. Jacob M. Montgomery Quantitative Political Methodology (L32 363) September 14, 2016 Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
2 Sampling Distributions Sampling Distributions A sampling distribution is the distribution of a statistic given repeated sampling. Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
3 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, / 6
4 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, / 6
5 Central limit theorem Central limit theorem For random sampling with a large sample size n, the sampling distribution ) σ of the sample mean ȳ is approximately normal, where ȳ N (µ, n. Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
6 Central limit theorem Central limit theorem For random sampling with a large sample size n, the sampling distribution ) σ of the sample mean ȳ is approximately normal, where ȳ N (µ, n. Some notes: As n, the standard error is going to get smaller and smaller. Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
7 Central limit theorem Central limit theorem For random sampling with a large sample size n, the sampling distribution ) σ of the sample mean ȳ is approximately normal, where ȳ N (µ, n. 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, / 6
8 Central limit theorem Central limit theorem For random sampling with a large sample size n, the sampling distribution ) σ of the sample mean ȳ is approximately normal, where ȳ N (µ, n. 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, / 6
9 It works for EVERYTHING Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
10 Sampling Distribution of ȳ A key common statistic is ȳ = 1 n y i for a single sample, where n is the sample size. How is this statistic distributed? Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
11 Sampling Distribution of ȳ A key common statistic is ȳ = 1 n y i for a single sample, where n is the 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, / 6
12 Sampling Distribution of ȳ A key common statistic is ȳ = 1 n y i for a single sample, where n is the 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, / 6
13 Sampling Distribution of ȳ A key common statistic is ȳ = 1 n y i for a single sample, where n is the 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 called the standard error of ȳ, and is equal to σ n. Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
14 Sampling Distribution of ȳ A key common statistic is ȳ = 1 n y i for a single sample, where n is the 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 called the standard error of ȳ, and is equal to σ n. Under certain circumstances we can safely assume that ȳ N(µ, σ n ). Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
15 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, / 6
16 Class business Attendance Lecture 5 (QPM 2016) Sampling Distributions September 14, / 6
17 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, / 6
18 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, / 6
Dummies and Interactions
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Class business PS is due Wed. Lecture 20 (QPM 2016) Multivariate Regression November 14, / 44
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