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Probability and Confidence Intervals in SAT Scores and Salaries

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viveka
21 diapositives · 331 Views · Sep 25, 2014

This practice guide covers statistical concepts, focusing on probability calculations for SAT scores and constructing confidence intervals for salaries. Examples include determining the likelihood of a sample mean SAT score being below a certain threshold, as well as deriving the confidence interval for the average salary of women compared to men with similar experience levels. Familiarity with Z-scores, standard errors, and t-distributions is essential for understanding these statistical techniques.

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Probability and Confidence Intervals in SAT Scores and Salaries
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  1. Practice • 8.17

  2. 8.17 • 110 or greater • Z = 105-100/3 = 1.67 p = .0475 • 90 or less • Z = 90-100/3 = -3.33 p = .0004

  3. Practice • For an SAT test •  = 500 •  = 100 • What is the probability that a sample of 65 people will have a mean SAT score below 525?

  4. Step 1: Sketch out question -3 -2 -1 0 1 2 3

  5. Step 2: Calculate the Standard Error 100 / 65 = 12.40 -3 -2 -1 0 1 2 3

  6. Step 3: Calculate the Z score (525 - 500) / 12.40 = 2.02 -3 -2 -1 0 1 2 3

  7. Step 4: Look up Z score in Table Z = 2.02; Column B =.4783 .50 .4783 -3 -2 -1 0 1 2 3

  8. Practice • There is a .9783 probability that a sample of 65 people would have a mean SAT under 525

  9. Practice • In a large corporation, the mean salary for all males with 3 to 5 years of experience was $28,000. Salaries (expressed in thousands) for a random sample of 10 women also having 3 to 5 years of experience were: • 24, 27, 31, 21, 19, 26, 30, 22, 15, 36 • Construct the 95% confidence interval for women and interpret what this means relative to the mean salary of males.

  10. Practice • M = 25.1 • SE = 1.97 • t(9) = 2.262 • 20.64 to 29.56 • 20,640 to 29,560

  11. Practice 8.30

  12. Practice • SE = 1.0 • t = 2.064 • T1 = 5.936 to 10.064 • T2 = 3.936 to 8.064 • T3 = 11.936 to 16.064 • T4 = 13.936 to 18.064

  13. Practice 8.23

  14. Practice • M= 27 • S hat = 3.803 • SE = 1.016 • t (13)= 2.160 • LL = 24.8 • UL = 29.2 • The workshop is working. We are 95% confident the average score of everyone who takes the workshop would be above 24 (the norm of the test)

  15. Practice • 8.12 • 8.13

  16. As N (sample size) increases the standard error decreases!

  17. Practice Handout

  18. Cookbook • Bring your cookbook to class on Friday!