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Quantitative Methods in Social Sciences (E774): Review Session V

Quantitative Methods in Social Sciences (E774): Review Session V. Dany Jaimovich October 21, 2009. Plan for today. Probability Distributions: Discrete and continuous variables Mean and Variance Normal Probability Distribution: Z-score PP1. Probability Distributions.

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Quantitative Methods in Social Sciences (E774): Review Session V

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  1. Quantitative Methods inSocial Sciences (E774): Review Session V Dany Jaimovich October 21, 2009

  2. Plan for today • Probability Distributions: • Discrete and continuous variables • Mean and Variance • Normal Probability Distribution: • Z-score • PP1

  3. Probability Distributions • Probability: expected proportion of realizations of a certain random variable. • 0≤P≤1 • The histogram of discrete variable shows its probability distribution:

  4. Probability Distributions

  5. Probability Distributions

  6. Probability Distributions MEAN OF A PD μ = 0*0.19+1*0.18+2*0.21+3*0.17+4*0.11+5*0.07+6*0.04+7*0.03 = 2.35

  7. Probability Distributions VARIANCE OF A PD σ² = (0-2.35)²*0.19+ (1-2.35)²*0.18+ (2-2.35) ²*0.21+ (3-2.35)²*0.17+ (4-2.35)²*0.11+ (5-2.35)²*0.07+ (6-2.35)²*0.04+ (7-2.35) ²*0.03 = 3.45 σ=1.86

  8. Probability Distributions • For continuous variables: • The probability is given by the area under the curve.

  9. Probability Distributions • Normal Probability Distribution:

  10. Probability Distributions • If μ for male age in the Geneva is 37 years old. • If σ is 10, • And age follows a Normal Probability Distribution, then taking a random guy in the street: • There are 68% of chances that he will between 27 and 47 years old. • There is 95% of chances that he will be between 17 and 57. • There is 99.7% chances that he will be between 7 and 67 years old.

  11. Probability Distributions • Generalizing this property of the Normal Probability Distribution, it is possible to find the likelihood of any value (x), and start asking more sophisticated research questions. • In order to do this, we need to “standardize” the values so we can be measure in terms of σ. This is the STANDARD SCORE or Z-SCORE:

  12. Probability Distributions • The values of z are known and are tabulated in a table (z-table). EXAMPLE 1 • In the previous example, μ =37, σ =10. • How likely is that the first guy I meet in the street is less than 51.5 years old?: • Z = (51.5-37)/10=1.45

  13. Probability Distributions

  14. Probability Distributions • How to read this???? • Then, the probability to find someone below 51.1 years old is 93%. • The accumulated probability between 0 and zis 43%. • The accumulated probability until 50%.

  15. Probability Distributions EXAMPLE 2 • Which is the probability to find someone between 51.1 and 27 years old? • Z(27) = (27-37)/10=-1. • Value from the table= 0.341 • Z(51.1) = 1.45. Table= 0.4265 • Probability ≈ 34%+43% ≈ 77%

  16. Probability Distributions EXAMPLE 3 • Which is the probability to find someone between 32 and 27 years old? • Z(32)= (32-37)/10=-0.5. • Value from the table= 0.1915 • Z(27) = -1. Table= 0.341 • Probability ≈ 34%-19% ≈15%

  17. Probability Distributions • One tail questions: Is x bigger – smaller than certain value? • Critical Z value at 95%≈1.65

  18. Probability Distributions • One tail questions: In Exercise 1, can we tell at 95% confidence that a random guy from street will be less thah 51.1? YES, Z<1.65

  19. Probability Distributions • Two tails questions: Is x equal to certain value? • Critical Z value at 95%=1.96

  20. Probability Distributions • Two tails questions: Is the age of the guy 100 at 95% confidence? • Critical Z (100)>1.96…. NO

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