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Economics 105: Statistics

Economics 105: Statistics. Review #1 to be handed out Tuesday, due following Tuesday in class. Take-home, closed-book, closed-notes, untimed, must use Excel or calculator (and transfer answers to the exam paper).

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Economics 105: Statistics

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  1. Economics 105: Statistics Review #1 to be handed out Tuesday, due following Tuesday in class. Take-home, closed-book, closed-notes, untimed, must use Excel or calculator (and transfer answers to the exam paper). Formula sheet rules: No words, in English or otherwise. Only formulas/equations. No proofs. Symbols like B for Binomial are okay. Front & back of 1 sheet of paper. Excel help is okay. Equation editor can be useful Go over GH 6, GH 7 & 8 due Tuesday

  2. Probability Distributions Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Bernoulli Normal Binomial Uniform Poisson Exponential Hypergeometric

  3. Exponential Distribution Graph Useful for waiting time, duration, or queuing problems Memoryless property Find the prob no student arrives in next hour. Find prob a student arrives in next 5 minutes.

  4. Probability Distributions Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Bernoulli Normal Binomial Uniform Poisson Exponential Hypergeometric

  5. Normal Distribution Let The p.d.f. is given by “The bell curve”, also sometimes called the Gaussian distribution after this guy http://cnx.rice.edu/content/m11161/latest/#java Reading the table … pages 915 in BLK, 11th edition. Note that numbers across the top (i.e., at top of each column) are the SECOND digit after the decimal.

  6. The Normal Distribution • ‘Bell Shaped’ • Symmetrical • Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to   f(X) σ X μ Mean = Median = Mode

  7. Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions

  8. StandardizedNormal Distribution The Z distribution always has mean = 0 and standard deviation = 1 f(Z) 1 Z 0 Values above the mean have positive Z-values, values below the mean have negative Z-values

  9. Example Convention If X ~ N(100, 2500), then the Z value for X = 200is This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

  10. Comparing X and Z units 100 200 X (μ = 100, σ = 50) 0 2.0 Z (μ = 0, σ = 1) Note that the distribution is the same, only the scale has changed.

  11. Finding Normal Probabilities Probability is measured by the area under the curve f(X) ) ≤ ≤ P ( a X b ) < < = P ( a X b (Note that the probability of any individual value is zero) X a b

  12. Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(X) 0.5 0.5 μ X

  13. Empirical Rules μ ± 1σ encloses about 68% of X’s What can we say about the distribution of values around the mean? There are some general rules: f(X) σ σ X μ-1σ μ μ+1σ 68.26%

  14. The Empirical Rule (continued) • μ ± 2σ covers about 95% of X’s • μ ± 3σ covers about 99.7% of X’s 3σ 3σ 2σ 2σ μ x μ x 95.44% 99.73%

  15. The Standardized Normal Table The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z) 0.9772 Example: P(Z < 2.00) = 0.9772 Z 0 2.00

  16. The Standardized Normal Table The value within the table gives the probability from Z =   up to the desired Z value (continued) The columngives the value of Z to the second decimal point Z 0.00 0.01 0.02 … 0.0 0.1 The rowshows the value of Z to the first decimal point . . . 2.0 .9772 P(Z < 2.00) = 0.9772 2.0

  17. Finding Normal Probabilities • Suppose X ~ N(8, 25). Find P(X < 8.6) X 8.0 8.6

  18. Suppose X ~ N(8, 25). Find P(X < 8.6) Finding Normal Probabilities (continued) μ = 8 σ = 10 μ= 0 σ = 1 X Z 8 8.6 0 0.12 P(X < 8.6) P(Z < 0.12)

  19. Solution: Finding P(Z < 0.12) Standardized Normal Probability Table (Portion) P(X < 8.6) = P(Z < 0.12) .02 Z .00 .01 .5478 .5000 0.0 .5040 .5080 .5398 .5438 .5478 0.1 0.2 .5793 .5832 .5871 Z 0.00 0.3 .6179 .6217 .6255 0.12

  20. Finding the X value for a Known Probability Example: • Suppose X ~ N(8, 25) • Find the X value so that only 20% of all values are below this X 0.2000 X ? 8.0 Z ? 0

  21. Find the Z value for 20% in the Lower Tail 1. Find the Z value for the known probability • 20% area in the lower tail is consistent with a Z value of -0.84 Standardized Normal Probability Table (Portion) .04 Z .03 .05 … -0.9 .1762 .1736 .1711 … 0.2000 .2033 .2005 .1977 -0.8 … … -0.7 .2327 .2296 .2266 X ? 8.0 Z -0.84 0

  22. Finding the X value 2. Convert to X units using the formula: So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80

  23. More Examples • If Z ~ N(0,1), find P(-1 < Z < 1) • If W ~ N(3,4), find P(-1 < W < 1)

  24. Evaluating Normality • Construct charts or graphs • For small- or moderate-sized data sets, do stem-and-leaf display and box-and-whisker plot look symmetric? • For large data sets, does the histogram or polygon appear bell-shaped? • Compute descriptive summary measures • Do the mean, median and mode have similar values? • Is the interquartile range approximately 1.33 σ? • Is the range approximately 6 σ?

  25. (continued) Evaluating Normality • Observe the distribution of the data set • Do approximately 2/3 of the observations lie within mean 1 standard deviation? • Do approximately 80% of the observations lie within mean 1.28 standard deviations? • Do approximately 95% of the observations lie within mean 2 standard deviations? • Evaluate normal probability plot • Is the normal probability plot approximately linear with positive slope?

  26. The Normal Probability Plot • Normal probability plot • Arrange data into ordered array • Find corresponding standardized normal quantile values • Plot the pairs of points with observed data values on the vertical axis and the standardized normal quantile values on the horizontal axis • Evaluate the plot for evidence of linearity

  27. The Normal Probability Plot (continued) A normal probability plot for data from a normal distribution will be approximately linear: X 90 60 30 Z -2 -1 0 1 2

  28. The Normal Probability Plot

  29. Normal Probability Plot (continued) Left-Skewed Right-Skewed X X 90 90 60 60 30 30 Z Z -1 -1 -2 0 1 2 -2 0 1 2 Rectangular Nonlinear plots indicate a deviation from normality X 90 60 30 Z -1 -2 0 1 2

  30. Other Continuous Distributions Source: wikipedia pages

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