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Probability and Confidence Intervals in SAT Scores and Salaries

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!

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