1 / 49

Week 7

Week 7. Sample Means & Proportions. Variability of Summary Statistics. Variability in shape of distn of sample Variability in summary statistics Mean, median, st devn, upper quartile, … Summary statistics have distributions. Parameters and statistics.

oneida
Télécharger la présentation

Week 7

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. Week 7 Sample Means & Proportions

  2. Variability of Summary Statistics • Variability in shape of distn of sample • Variability in summary statistics • Mean, median, st devn, upper quartile, … • Summary statistics have distributions

  3. Parameters and statistics • Parameter describes underlying population • Constant • Greek letter (e.g. , , , …) • Unknown value in practice • Summary statistic • Random • Roman letter (e.g. m, s, p, …) • We hope statistic will tell us about corresponding parameter

  4. Distn of sample vsSampling distn of statistic • Values in a single random sample have a distribution • Single sample --> single value for statistic • Sample-to-sample variability of statistic is its sampling distribution.

  5. Means • Unknown population mean,  • Sample mean, X, has a distribution — its sampling distribution. • Usually x ≠  • A single sample mean, x, gives us information about 

  6. Sampling distribution of mean If sample size, n, increases: • Spread of distn of sample is (approx) same. • Spread of sampling distn of mean gets smaller. • x is likely to be closer to  • x becomes a better estimate of 

  7. Sampling distribution of mean Population with mean , st devn  Sample mean, X, has sampling distn with: • Mean, • St devn, Random sample (n independent values) (We will deal later with the problem that  and  are unknown in practice.)

  8. Weight loss Estimate mean weight loss for those attending clinic for 10 weeks • Random sample of n = 25 people • Sample mean, x How accurate? Let’s see, if the population distn of weight loss is:

  9. Some samples Four random samples of n = 25 people: • Mean = 8.32 pounds, st devn = 4.74 pounds • Mean = 8.32 pounds, st devn = 4.74 pounds • Mean = 8.48 pounds, st devn = 5.27 pounds • Mean = 7.16 pounds, st devn = 5.93 pounds N.B. In all samples, x ≠ 

  10. Sampling distribution Means from simulation of 400 samples Theory: mean =  = 8 lb, s.d.( ) = lb (How does this compare to simulation? To popn distn?)

  11. Errors in estimation Population • From 70-95-100 rule • x will be almost certainly within 8 ± 3 lb • x is unlikely to be more than 3 lb in error • Even if we didn’t know  • x is unlikely to be more than 3 lb in error Sampling distribution of mean mean =  = 8 lb, s.d.( ) = lb

  12. Increasing sample size, n If we sample n = 100 people instead of 25: s.d.( ) = lb. Larger samples more accurate estimates

  13. Central Limit Theorem • If population is normal (, ) • If popn is non-normal with (, ) but n is large Guideline: n > 30 even if very non-normal

  14. Other summary statistics E.g. Lower quartile, proportion, correlation • Usually not normal distns • Formula for standard devn of samling distn sometimes • Sampling distn usually close to normal if n is large

  15. Lottery problem Pennsylvania Cash 5 lottery • 5 numbers selected from 1-39 • Pick birthdays of family members (none 32-39) • P(highest selected is 32 or over)? Statistic: H = highest of 5 random numbers (without replacement)

  16. Lottery simulation Theory? Fairly hard. Simulation: Generated 5 numbers (without replacement) 1560 times Highest number > 31 in about 72% of repetitions

  17. Normal distributions • Family of distributions (populations) • Shape depends only on parameters (mean) &  (st devn) • All have same symmetric ‘bell shape’ • = 65 inches, s = 2.7 inches

  18. Importance of normal distn • A reasonable model for many data sets • Transformed data often approx normal • Sample means (and many other statistics) are approx normal.

  19. Standard normal distribution • Z ~ Normal ( = 0,  = 1) -3 -2 -1 0 1 2 3 • Prob ( Z < z* )

  20. Probabilities for normal (0, 1) P(Z  -3.00) = P(Z  −2.59) = P(Z  1.31) = P(Z  2.00) = P(Z  -4.75) = 0.0013 0 .0048 0 .9049 0 .9772 0 .000001 Check from tables:

  21. Probability Z > 1.31 P(Z > 1.31) = 1 – P(Z  1.31) = 1 – .9049 = .0951

  22. Prob ( Z between –2.59 and 1.31) P(-2.59 Z  1.31) = P(Z  1.31) – P(Z  -2.59) = .9049 – .0048 = .9001

  23. Standard devns from mean • Normal (, )     • Heightsof students • = 65 inches, s = 2.7 inches

  24. Probability and area X ~ normal ( = 65 , s = 2.7 ) P (X ≤ 67.7) = area

  25. Probability and area (cont.) Exactly70-95-100 rule • P(X within  of ) = 0.683 approx 70% • P(X within 2 of ) = 0.954 approx 95% • P(X within 3 of ) = 0.997 approx 100% • Normal (, )    

  26. Finding approx probabilities Ht of college woman, X ~ normal ( = 65 , s = 2.7 ) Prob (X ≤ 62 )? Sketch normal density Estimate area P (X ≤ 62) = area About 1/8

  27. Translate question from X to Z Translate to z-score: • Z ~ Normal ( = 0,  = 1) • X ~ Normal (, ) • Find P(X ≤ x*)     x* -3 -2 -1 0 1 2 3 z*

  28. Finding probabilities Prob (height of randomly selected college woman ≤ 62 )? About 13%.

  29. Prob (X > value) Ht of college woman, X ~ normal ( = 65 , s = 2.7 ) Prob (X > 68 inches)?

  30. Finding upper quartile Blood Pressures are normal with mean 120 and standard deviation 10. What is the 75th percentile? Step 1: Solve for z-score Closest z* with area of 0.7500 (tables) z = 0.67 Step 2: Calculate x = z*s+ m x = (0.67)(10) + 120 = 126.7 or about 127.

  31. Probabilities about means • Blood pressure ~ normal ( = 120,  = 10) • 8 people given drug • If drug does not affect blood pressure, • Find P(average blood pressure > 130)

  32. P ( X > 130) ? • prob = 0.0023 • X ~ normal ( = 120,  = 10) • n = 8 Very little chance!

  33. Distribution of sum X ~ distn with (, )  aX ~ distn with (a, a) e.g. milesto kilometers  Central Limit Theorem implies approx normal

  34. Probabilities about sum • Profit in 1 day ~ normal (= $300, = $200) • Prob(total profit in week < $1,000)? • Total = • Prob = 0.0188 Assumes independence

  35. Categorical data • Most important parameter is •  = Prob (success) • Corresponding summary statistic is • p = Proportion (success) ^ N.B. Textbook uses p and p

  36. Number of successes • Easiest to deal with count of successes before proportion. • If… • 1. n “trials” (fixed beforehand). • 2. Only “success” or “failure” possible for each trial. • 3. Outcomes are independent. • Prob (success), remains same for all trials, . • Prob (failure) is 1 – . • X = number of successes ~ binomial (n, )

  37. Examples

  38. Binomial Probabilities for k = 0, 1, 2, …, n You won’t need to use this!! Prob (win game) = 0.2 Plays of game are independent. What is Prob (wins 2 out of 3 games)? What is P(X = 2)?

  39. Mean & st devn of Binomial For a binomial (n, )

  40. Extraterrestrial Life? 50% of large population would say “yes” if asked, “Do you believe there is extraterrestrial life?” Sample of n = 100 X = # “yes” ~ binomial (n = 100,  = 0.5)

  41. Extraterrestrial Life? Sample of n = 100 X = # “yes” ~ binomial (n = 100,  = 0.5) 70-95-100 rule of thumb for # “yes” • About 95% chance of between 40 & 60 • Almost certainly between 35 & 65

  42. Normal approx to binomial If X is binomial (n , ), and n is large, then X is also approximately normal, with Conditions: Both nand n(1 – ) are at least 10. (Justified by Central Limit Theorem)

  43. Number of H in 30 Flips X = # heads in n = 30 flips of fair coinX ~ binomial ( n = 30, = 0.5) Bell-shaped & approx normal.

  44. Opinion poll n = 500 adults; 240 agreed with statement If  = 0.5 of all adults agree, what P(X ≤ 240) ? X is approx normal with Not unlikely to see 48% or less, even if 50% in population agree.

  45. Sample Proportion • Suppose (unknown to us) 40% of a population carry the gene for a disease, ( = 0.40). • Random sample of 25 people; X = # with gene. • X ~ binomial (n = 25 ,  = 0.4) p = proportion with gene

  46. Distn of sample proportion • X ~ binomial (n , ) Large n: p is approx normal (n ≥ 10 &n (1 – ) ≥ 10)

  47. Examples • Election Polls:to estimate proportion who favor a candidate; units = all voters. • Television Ratings:to estimate proportion of households watching TV program; units = all households with TV. • Consumer Preferences:to estimate proportion of consumers who prefer new recipe compared with old; units = all consumers. • Testing ESP:to estimate probability a person can successfully guess which of 5 symbols on a hidden card; repeatable situation = a guess.

  48. Public opinion poll Suppose 40% of all voters favor Candidate A. Pollsters sample n = 2400 voters. Propn voting for A is approx normal Simulation 400 times & theory.

  49. Probability from normal approx If 40% of voters favor Candidate A, and n = 2400 sampled Sample proportion, p, is almost certain to be between 0.37 and 0.43 Prob 0.95 of p being between 0.38 and 0.42

More Related