Chapter 8 From Samples to Populations Sampling Distributions

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 p 209, Central Limit Theorem) GED 111/CDS 111 Statistics in Modern Society 2

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. GED 111/CDS 111 Statistics in Modern Society 3

Exercise • Q 5 -8 p 343 GED 111/CDS 111 Statistics in Modern Society 4

Estimating Population Means • Point estimate – Sample mean • Interval Estimate – Confidence Interval – 95% confidence interval for a population mean – Margin of error, E, ≈ 2 s/√n • S is the standard deviation of the sample • n is the sample size GED 111/CDS 111 Statistics in Modern Society 5

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 p 348 GED 111/CDS 111 Statistics in Modern Society 6

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) GED 111/CDS 111 Statistics in Modern Society 7

Exercise • Q 5 -8 p 353 GED 111/CDS 111 Statistics in Modern Society 8

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 GED 111/CDS 111 Statistics in Modern Society 9

Choosing Sample Size to estimate P • At least • n = 1/E 2 • Any sample size equal to or larger than this value will suffice GED 111/CDS 111 Statistics in Modern Society 10

• Exercise – Q 22, 30 p 359 -360 • Focus on Literature – How Many Words Did Shakespeare Know? P 366 -367 GED 111/CDS 111 Statistics in Modern Society 11