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Chapter 7

Chapter 7 . Confidence Intervals and Sample Size. 7.2 Estimators. 7.2 Interval Estimates. Does the interval actually contain the parameter?. Based on the desired confidence level, we set our confidence interval:. 7.2 Confidence Intervals. σ known or n ≥30.

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Chapter 7

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  1. Chapter 7 Confidence Intervals and Sample Size

  2. 7.2 Estimators

  3. 7.2 Interval Estimates Does the interval actually contain the parameter? Based on the desired confidence level, we set our confidence interval:

  4. 7.2 Confidence Intervals σ known or n ≥30 The three most commonly used confidence intervals are: 99% 95% 90% .5 -.025 = .475 1.65 2.58 .475 1.96

  5. 7.2 Confidence Intervals σ known or n ≥30 If we do not know σ, we can use s, so long as n ≥ 30. • Rounding Rule for Confidence Intervals: • If you are given raw data round to one more place than the original data. • If you are given a sample mean, round to the same number of places as the given mean.

  6. 7.2 Maximum Error Estimate Area in One Tail Outside of Interval Population Standard Deviation Sample Size

  7. 7.2 Example 1 n = 50 σ = 6.8

  8. 7.2 Example 1 σ = 6.8 n = 50 We are 95% confident that the true population mean is between 17.1 and 20.9

  9. X = 7.2 Example 2 n = σ=

  10. 7.2 Example 3

  11. 7.2 Example 4 Find the 98% confidence interval of population mean for the last example. .49 .5- .01 = .49

  12. 7.2 Example 4 Use z = 2.33

  13. 2.33 2.33 7.2 Example 4 5.6 – 2.33(.1461) < μ < 5.6 + 2.33(.1461) 5.6 – .3404 < μ < 5.6 + .3404 5.260 < μ < 5.940 5.3 < μ < 6.0

  14. 7.2 Sample Size √n √n E E

  15. 7.2 Example 5 σ = 3 We need a sample of 60 students

  16. 7.2 Example 6 E = σ =

  17. 7.3 Confidence Intervals σ unknown and n ≤ 30

  18. 7.3 Confidence Intervals σ unknown and n ≤ 30 Table F Page 771

  19. X Sample Standard Deviation 7.3 Confidence Intervals σ unknown and n ≤ 30 sSample Mean µ Population Mean n Sample Size

  20. 7. 3 Example 6 Find the degrees of Freedom Look for 21 in first column of Table F Page 771 Go over 5 columns to 95%

  21. 7.3Example 6

  22. 7.3 Example 7 n = 10 < 30 and σ is unknown 10 – 1 = 9 degrees of freedom X = .32 s = .08

  23. 7.3 Example 7 X =.32 s =.08 tα/2 =2.262

  24. 7.3 Example 8

  25. 7.3 When to use tα/2 verses zα/2

  26. 7.4 Proportions Proportions can be represented as percents, decimals or fractions. If in a class of 35 students, if 7 are commuters the proportion of commuters in the class is:

  27. 7.4 Example 9 or

  28. 7.4 Confidence Intervals Maximum Error Estimate Rounding Rule for Confidence Intervals for Proportions: Round to three decimal places.

  29. 7.4 Example 10

  30. 7.4 Sample Size for Proportions Solve for n

  31. 7.4 Example 11

  32. 7.4 Example 12

  33. 7.5 Confidence Intervals for σ and σ2 • Confidence Intervals for μ • n ≥ 30 or σ know; Normal Distribution • n < 30 and σ unknown; t-Distribution • Confidence Intervals Proportions • Normal Distribution • Confidence Intervals for σ and σ2 • Chi-Square Distribution Table G pg 772

  34. 7.5 Chi-square Distribution • Properties: • Variable cannot be negative • Positively skewed • Area under curve is 1 or 100% Symbol for Chi-square distribution

  35. 7.5 Confidence Intervals for σ and σ2 Our formulas require two values from the Chi-square Distribution

  36. 7.5 Left and Right Chi Values α = 1 –Confidence Level 100 α/2 1 - α/2

  37. 7.5 Example 13 2 Degrees freedom = n – 1 = 25 – 1 = 24

  38. 7.5 Example 13 Use left column .95, right column .05, and row 24

  39. 7.5 Example 14 Right Left α =1 - .95 = .05 1 – α/2 = 1 - .025 = .975 α/2 =.05/2 = .025 Degrees freedom = 20 -1 = 19

  40. 7.5 Example 14

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