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Chapter 7

“Estimating Population Parameters”. Chapter 7. Chapter 7 Summary By Chris Schulze. Confidence Intervals. 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.

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Chapter 7

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  1. “Estimating Population Parameters” Chapter 7 Gonick: The Cartoon Guide To Statistics (Chapter 7) Chapter 7 Summary By Chris Schulze Confidence Intervals

  2. Statistical Inferences • Given a single observation how do we use p and X to evaluate it? Gonick: The Cartoon Guide To Statistics (Chapter 7)

  3. 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)

  4. 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)

  5. 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)

  6. 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: • You can claim, with 95% confidence, that the probability of success in the full population is in the interval p ± 2SEp. • The size of the standard error depends on the sample SE (p) = p(1-p) n SE = Standard Error n = Sample Size p = Probability of Success Gonick: The Cartoon Guide To Statistics (Chapter 7) .95 ≈ Pr (p – 1.96SE(p) ≤ p ≤ p + 1.96SE(p))

  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: • You can claim with 95% confidence that the mean of the full population is in the interval x ± 2SEx. • The standard error depends on the sample size • The confidence interval can be made smaller by increasing the sample size. SE (X) = s n s = standard deviation of the sample n = sample size Gonick: The Cartoon Guide To Statistics (Chapter 7) .95 ≈ Xr (x – 1.96SE(x) ≤ x ≤ x + 1.96SE(x))

  8. Estimating the Mean (Example) • The equation: • 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) = 66.4890 + (1.96)(12) = 113.52 SE (X) = s n Gonick: The Cartoon Guide To Statistics (Chapter 7)

  9. 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)

  10. 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)

  11. 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)

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