1 / 29

Economics 105: Statistics

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

beck
Download Presentation

Economics 105: Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Economics 105: Statistics Review #1 to be handed out Monday, due next Monday 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 7, GH 8 is due Monday

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

  3. 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.

  4. 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

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

  6. Normal Distribution Shape f(X) Changingμ shifts the distribution left or right. Changing σ increases or decreases the spread. σ μ X

  7. Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation:

  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 If X is distributed normally with mean of 100 and standard deviation of 50, 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) • The number of car accidents in a city in a given month is normally distributed. Thirty-three percent of the time more than 200 occur. Fewer than 50 occur 1.5 percent of the time. What are the mean and standard deviation of the number of accidents in a month?

  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

More Related