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CHAPTER 7 Confidence Intervals “Estimating Population Parameters” Chapter 7 Summary By Chris Schulze Gonick: The Cartoon Guide To Statistics (Chapter 7) STATISTICAL INFERENCES Given a single observation how do we use p and X to evaluate it? Gonick: The Cartoon Guide To Statistics (Chapter 7) INDUCTION Inductive Reasoning is reasoning from a specific case to a general rule. Going from a set of observations to a hypothesis. Gonick: The Cartoon Guide To Statistics (Chapter 7) CHAPTER 7 – CONFIDENCE INTERVALS It is a range of values used to estimate the true value of a population parameter. By providing a range of values likely to contain the population parameter of interest, confidence intervals help to determine how well the sample statistic estimates the underlying population value. The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter. Gonick: The Cartoon Guide To Statistics (Chapter 7) CONFIDENCE INTERVAL To estimate the probability of success in the full population, you can report a range around the probability of success found in the sample. The size of the range depends on the confidence level 95% is commonly used Other common confidence intervals are: 90% and 99%. Gonick: The Cartoon Guide To Statistics (Chapter 7) CONFIDENCE INTERVAL (CONT.) A 95% confidence interval is created by using the “two standard deviations” rule except that you use standard error instead of the standard deviation. The standard error is defined as: SE (p) = p(1-p) n SE = Standard Error n = Sample Size p = Probability of Success You can claim, with 95% confidence, that the probability of success in the full population is in the interval p ± 2SEp. .95 ≈ Pr (p – 1.96SE(p) ≤ p ≤ p + 1.96SE(p)) The size of the standard error depends on the sample Gonick: The Cartoon Guide To Statistics (Chapter 7) ESTIMATING THE MEAN To estimate the mean of some full population, you can report a range around the mean ¯x of the sample. Again, the size of the range depends on the desired confidence level. For a 95% confidence interval you again use twice the standard error where the standard error is defined as: s = standard deviation of the sample n = sample size SE (X) = s n You can claim with 95% confidence that the mean of the full population is in the interval x ± 2SEx. .95 ≈ Xr (x – 1.96SE(x) ≤ x ≤ x + 1.96SE(x)) The standard error depends on the sample size The confidence interval can be made smaller by increasing the sample size. Gonick: The Cartoon Guide To Statistics (Chapter 7) ESTIMATING THE MEAN (EXAMPLE) The equation: SE (X) = s n Example: Mean of 90 and standard deviation of 12. The figure shows the distribution The shaded area is the middle 95% of the distribution (66.48-113.52) 90 - (1.96)(12) = 66.48 90 + (1.96)(12) = 113.52 Gonick: The Cartoon Guide To Statistics (Chapter 7) KEY TERMS TO REMEMBER Induction Confidence Interval Estimating the Mean Standard Error 90%, 95%, 99% confidence interval Gonick: The Cartoon Guide To Statistics (Chapter 7) Helpful Websites http://www.dimensionresearch.com/index.html http://stattrek.com http://easycalculation.com/statistics http://www.tutorvista.com http://www.stat.yale.edu/Courses/1997-98/101/fonfint.htm http://mathworld.wolfram.com/ConfidenceInterval.html Gonick: The Cartoon Guide To Statistics (Chapter 7) REFERENCES http://www.wikipedia.org/ http://www.khanacademy.org/ http://onlinestatbook.com/chapter8/mean.html http://www.stat.yale.edu/Courses/1997-98/101/confint.htm The Cartoon Guide to Statistics, Gonick, 1993 An Introduction to Statistical Problem Solving in Geography, McGrew, 1993 Gonick: The Cartoon Guide To Statistics (Chapter 7)