Download Confidence Intervals for Population Means: Concepts, Calculation, and Interpretation - Pro and more Study notes Statistical mechanics in PDF only on Docsity! 3/31/2009 1 L E C T U R E 1 8 STA291 Fall 2009 1 Preview & Administrative Notes 2 • 10 Estimation – 10.1 Concepts of Estimation • • Suggested Reading – Study Tools Chapter 10.1, 10.2 – OR: Sections 10.1, 10.2 in the textbook • Suggested problems from the textbook: 10.1, 10.2, 10.6, 10.10, 10.12, 10.14, 10.16 10.41, 10.42, 10.51, 12.54, 12.55, 12.58, 12.65 • Le Menu • 10 Estimation – 10.1 Concepts of Estimation – 10.2 Estimating the Population Mean – 10.3 Selecting the Sample Size 3 – (12.3) Confidence Interval for a Proportion 3/31/2009 2 Confidence Intervals 4 • A large-sample 95% confidence interval for the population mean is 96.1 n sX ± • where is the sample mean and • s is the sample standard deviation X Confidence Intervals—Interpretation 5 • “Probability” means that “in the long run, 95% of these intervals would contain the parameter” • If we repeatedly took random samples using the same method, then, in the long run, in 95% of the cases, th fid i t l ill (i l d ) th t e con ence n erva w cover nc u e e rue unknown parameter • For one given sample, we do not know whether the confidence interval covers the true parameter • The 95% probability only refers to the method that we use, but not to the individual sample Confidence Intervals—Interpretation 6 3/31/2009 5 Different Confidence Coefficients 13 Confidence Coefficient α α/2 zα/2 .90 .10 95 1 96. . .98 .99 2.58 3.00 Facts about Confidence Intervals 14 • The width of a confidence interval – ________ as the confidence coefficient increases – ________ as the error probability decreases – ________ as the standard error increases – ________ as the sample size increases Facts about Confidence Intervals II 15 • If you calculate a 95% confidence interval, say from 10 to 14, there is no probability associated with the true unknown parameter being in the interval or not Th t t i ith i th i t l f t • e rue parame er s e er n e n erva rom 10 o 14, or not – we just don’t know it • The 95% refers to the method: If you repeatedly calculate confidence intervals with the same method, then 95% of them will contain the true parameter 3/31/2009 6 Choice of Sample Size 16 • So far, we have calculated confidence intervals starting with z, s, n: • These three numbers determine the margin of error of the confidence interval: n szX ± sz • What if we reverse the equation: we specify a desired precision B (bound on the margin of error)??? • Given z and s, we can find the minimal sample size needed for this precision n Choice of Sample Size 17 • We start with the version of the margin of error that includes the population standard deviation, σ, setting that equal to B: n zB σ= • We then solve this for n: , where means “round up”. ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 2 2 2 B zn σ ⎡ ⎤ Example 18 • For a random sample of 100 UK employees, the mean distance to work is 3.3 miles and the standard deviation is 2.0 miles. • Find and interpret a 90% confidence interval for the id ti l di t f k f ll UK mean res en a s ance rom wor o a employees. • About how large a sample would have been adequate if we needed to estimate the mean to within 0.1, with 90% confidence?
