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Comparing Means and Proportions: Understanding Differences in Weight Loss and Other Factors

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This material explores the concepts of comparing two means or proportions through independent samples. It covers crucial statistical measures such as population parameters and estimates, confidence intervals, and hypothesis testing. Key examples include examining average weight loss between dieters and those who exercise, and differences in smoking cessation rates between treatments like bupropion and nicotine patches. Further discussions include methods for estimating differences and importance of random sampling in studies, enhancing understanding for making informed decisions based on statistical evidence.

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Comparing Means and Proportions: Understanding Differences in Weight Loss and Other Factors

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  1. Week 10 Comparing Two Means or Proportions

  2. Generalising from sample

  3. Generalising from sample

  4. Numerical measurements: means • Difference in average weight loss for those who diet compared to those who exercise to lose weight? • Difference is there between the mean foot lengths of men and women? • Population parameter • 2 – 1 = difference between population means • Sample estimate • x2 – x1 = difference between sample means

  5. Categorical measurements: propns • Difference between the proportions that would quit smoking if taking the antidepressant buproprion (Zyban) versus wearing a nicotine patch? • Difference between proportion who have heart disease of men who snore and men who don’t snore? • Population parameter • 2 – 1 = difference between population proportions • Sample estimate • p2 – p1 = difference between sample proportions

  6. Requirement: independent samples Two samples are called independent samples when the measurements in one sample are not related to the measurements in the other sample. • Random samples taken separately from 2 populations • Randomised experiment with 2 treatments • One random sample, but a categorical variable splits individuals into 2 groups.

  7. Model for numerical data • Sample 1 ~ population (mean 1, s.d. 1)Sample 2 ~ population (mean 2, s.d. 2) • Estimation: estimate (2 – 1) with • Standard error? • Confidence interval? • Testing: is (2 – 1) zero? • p-value

  8. Model for categorical data • Sample 1 ~ population (proportion 1)Sample 2 ~ population (proportion 2) • Estimation: estimate (2 – 1) with (p2 – p1) • Standard error? • Confidence interval? • Testing: is (2 – 1) zero? • p-value

  9. Distribution of difference • In both cases, we need to find distribution of difference • (p2 – p1) or • Independent samples >> difference of independent random variables. • We already know distns of the two parts — what is distn of their difference?

  10. Sum of 2 variables Same distns • Sample mean: • Sample total: Different distns

  11. Difference between 2 variables • Same standard devn as sum • If X1and X2 are normal • Remember that X1and X2 must be independent

  12. Example • Husband height ~ normal(1.85, 0.1)Wife height ~ normal(1.7, 0.08) • Assume independent. (Probably not!!) • Prob that wife is taller than husband? • (Husband - Wife) ~

  13. Example • Husband height ~ normal(1.85, 0.1)Wife height ~ normal(1.7, 0.08) • Husband - Wife ~ normal(0.15, 0.1281) P (diff ≤ 0) = area -0.11 0.02 0.15 0.28 0.41 Prob = 0.297

  14. Difference between proportions • If X1 and X2 are independent, • If p1 and p2 are independent, • For large samples, p1 and p2 are approx normal, so their difference is too.

  15. Std error for difference in propns Nicotine patches vs Antidepressant (Zyban)? n1 = n2 = 244 randomly assigned to each treatment Zyban: 85 out of 244 quit smokingPatch: 52 out of 244 quit smoking So,

  16. Approximate 95% C.I. For sufficiently large samples, the interval Estimate  2  Standard error is an approximate 95% C.I. • Best you can do for difference between proportions • For means, CI can be improved by replacing ‘2’ by a different value.

  17. Patch vs Antidepressant Study: n1 = n2 = 244 randomly assigned to each group Zyban: 85 of the 244 Zyban users quit smoking = .348 Patch: 52 of the 244 patch users quit smoking = .213 So, Approx 95% C.I. .135  2(.040) => .135  .080 => .055 to .215 We are 95% confident that Zyban gives an improvement of between 5.5% and 21.5% of the probability of quitting smoking.

  18. Difference between means • If X1 and X2 are independent, • If X1 and X2 are independent, • If both populations are normal, so is the difference.

  19. Std error for difference in means Lose More Weight by Diet or Exercise? n1 = 42 men on diet, n2 = 27 men on exercise routine Diet: Lost an average of 7.2 kg with std dev of 3.7 kgExercise: Lost an average of 4.0 kg with std dev of 3.9 kg So,

  20. Diet vs Exercise Study: n1 = 42 men on diet, n2 = 27 men exercise Diet: Lost an average of 7.2 kg with std dev of 3.7 kgExercise: Lost an average of 4.0 kg with std dev of 3.9 kg So, Approximate 95% Confidence Interval: 3.2  2(.81) => 3.2  1.62 => 1.58 to 4.82 kg We are 95% confident that those who diet lose on average 1.58 to 4.82 kg more than those who exercised.

  21. Better C.I. for mean A CI for the Difference Between Two Means(Independent Samples): where t*is a value from t-tables. • d.f. = min(n1–1, n2–1) • Welch’s approx gives a different d.f. (higher) but is a complicated formula • t* is approx 1.96 if d.f. is high

  22. Effect of a stare on driving Randomized experiment: Researchers either stared or did not stare at drivers stopped at a campus stop sign; Timed how long (sec) it took driver to proceed from sign to a mark on other side of the intersection. No Stare Group (n = 14): 8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8, 7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7 Stare Group (n = 13): 5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5, 6.1, 4.8, 4.9, 4.5, 7.2, 5.8 Estimate difference between the mean crossing times.

  23. Checking data • No outliers; no strong skewness. • Crossing times in stare group seem faster & less variable.

  24. Effect of stare on driving Using df = min(n1–1, n2–1) = 12, gives t* = 2.179 A 95% CI for 2–1 is

  25. Effect of stare on driving Minitab N.B. C.I. is based on df = 21 (Welch’s approx) • Slightly narrower C.I. that we got with d.f. = 12.

  26. Interpretation A 95% CI for 2–1 is 0.17 to 1.91 sec • We are 95% confident that it takes drivers between 0.17 and 1.91 seconds less on average to cross intersection if someone stares at them.

  27. Testing two proportions • Hypotheses H0: 1 – 2= 0 HA: 1 – 2≠ 0 or1 – 2< 0 or1 – 2> 0 Watch how Population 1 and 2 are defined. • Data requirements • Independent samples • n1 p1, n1(1-p1), n2 p2, n2(1-p2)all at least 5, preferably ≥10

  28. Test statistic • Based on p1 – p2 • Standardise:

  29. Test statistic • If H0 is true, best estimate of  is • So we use test statistic • If H0 is true, this has standard normal distn • p-value from normal distn

  30. Prevention of Ear Infections • Does the use of sweetener xylitol reduce the incidence of ear infections? Randomized Experiment: Of 165 children on placebo, 68 got ear infection. Of 159 children on xylitol, 46 got ear infection. • Hypotheses:H0: 1 – 2=0 Ha: 1 – 2>0 • Data check: At least 5 success & failure in each group

  31. Prevention of Ear Infections • Overall propn getting infection • Test statistic • p-value = 0.01 • Conclusion: • Strong evidence xylitol reduces chance of ear infection

  32. Testing two means • Hypotheses H0: 1 – 2= 0 HA: 1 – 2≠ 0 or1 – 2< 0 or1 – 2> 0 Watch how Population 1 and 2 are defined. • Data requirements • Fairly large n1 and n2 (say 30 or more), or • Not much skewness & no outliers (normal model reasonable)

  33. Test statistic • Based on • Standardise:

  34. Test • Test statistic: • If H0 is true, this has approx t-distn with d.f. = min(n1–1, n2–1) • Same d.f. as CI for 1 – 2 • p-value from t distn • Minitab or Excel • n1 and n2 ≥ 30 • Use normal tables

  35. Effect of a stare on driving Randomized experiment: Researchers either stared or did not stare at drivers stopped at a campus stop sign; Timed how long (sec) it took driver to proceed from sign to a mark on other side of the intersection. No Stare Group (n = 14): 8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8, 7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7 Stare Group (n = 13): 5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5, 6.1, 4.8, 4.9, 4.5, 7.2, 5.8 Test whether stare speeds up crossing times.

  36. Checking data • Small sample sizes, but • No outliers; no strong skewness.

  37. Effect of stare on driving H0: 1 – 2= 0 HA: 1 – 2> 0 where 1 = no-stare, 2 = stare • Hypotheses

  38. Effect of stare on driving • Test statistic • P-value df = min(n1–1, n2–1) = 12 Upper tail area of t-distn (12 d.f.) p = 0.016 Strong evidence that stare speeds up crossing

  39. Effect of stare on driving Minitab N.B. Test is based on df = 21 (Welch’s approx) • Very similar p-value and same conclusion Strong evidence that stare speeds up crossing

  40. Paired data and 2-sample data • Make sure you distinguish between: • 2 measurements on each individual (e.g. before & after) • Measurements from 2 independent groups • Different cars assessed for insurance claims in garages A and B • Same cars assessed by both garages 2 independent samples Paired data

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