From Samples to Populations

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From Samples to Populations Chapter 8 From Samples to Populations

Sampling Distributions The distribution of sample means is the distribution that results when we find the means of all possible samples of a given size. The larger the sample size, the more closely this distribution approximates a normal distribution. In all cases, the mean of the distribution of sample means equals the population mean. If only one sample is available, its sample mean is the best estimate for the population mean, µ. (Chapter 5 p209, Central Limit Theorem) GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Sampling Error The error that is introduced when a random sample is used to estimate a population parameter is called a sampling error. It does NOT include other sources of error, such as those due to biased sampling, bad survey questions, or recording mistakes. A practical way to reduce the sampling error would be to increase the sample size, n. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Q5-8 p343 GED111/CDS111 Statistics in Modern Society

Estimating Population Means Point estimate Sample mean Interval Estimate Confidence Interval 95% confidence interval for a population mean Margin of error, E, ≈ 2s/√n S is the standard deviation of the sample n is the sample size GED111/CDS111 Statistics in Modern Society

95% confidence interval for a population mean From (sample mean – margin of error) to (sample mean + margin of error) Sample mean – E < µ < sample mean + E Or Sample mean ± E Figure 8.10 p348 GED111/CDS111 Statistics in Modern Society

Choosing Sample Size to estimate µ n = (2σ/E)2 σ is the population standard deviation (usually estimated) Margin of error, E, is often known in advance (being specified) GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Q5-8 p353 GED111/CDS111 Statistics in Modern Society

95% confidence interval for a population proportion E ≈ 2√((p(1-p)/n) p is the sample proportion n is the sample size From p – margin of error to p + margin of error p – E < P < p + E ^ ^ ^ ^ ^ ^ ^ GED111/CDS111 Statistics in Modern Society

Choosing Sample Size to estimate P At least n = 1/E2 Any sample size equal to or larger than this value will suffice GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Q22, 30 p359-360 Focus on Literature How Many Words Did Shakespeare Know? P366-367 GED111/CDS111 Statistics in Modern Society