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Comparing Means: Reading Comprehension Scores of New vs. Traditional Programs

This chapter explores a study that compares the reading comprehension scores of elementary school children using a new reading activities program versus a traditional program. It discusses the process of comparing means, analyzing box plots, estimating parameters, and conducting hypothesis tests.

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Comparing Means: Reading Comprehension Scores of New vs. Traditional Programs

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  1. Chapter 24 Comparing Means

  2. Comparing Two Means • An educator believes that new reading activities for elementary school children will improve reading comprehension scores. She randomly assigns her third-grade students to one of two groups. The first group will use a traditional reading program and the second group will use the new reading activities. At the end of the experiment, both groups take a reading comprehension exam. Are the scores for the new reading activities group higher than for the traditional group?

  3. Comparing Two Means • Look at boxplot of each group’s scores.

  4. Comparing Two Means • What do you see? • ____________________________________ ____________________________________ • ____________________________________ ____________________________________

  5. Comparing Two Means • Does the new reading program produce better average scores? • For this particular class _________________________________ • For population of all third-graders _________________________________

  6. Comparing Two Means • μ1 = _____________________________ • μ2 = _____________________________ • Interested in quantity μ1 - μ2.

  7. Comparing Two Means • μ1 and μ2 are parameters (unknown). • ________________________________ • Estimate μ1 - μ2 with

  8. Sampling Distribution • Assumptions: • Random Samples • Samples are Independent • Nearly Normal Population Distributions

  9. Sampling Distribution • If Assumptions hold, sampling distribution is

  10. Sampling Distribution • σ1 and σ2 are parameters (unknown). • ____________________________ • ____________________________

  11. Sampling Distribution

  12. Degrees of Freedom? • t really doesn’t have a t distribution. • The true distribution of t is ________________________________ when you use this formula for the degrees of freedom.

  13. Degrees of Freedom?

  14. Degrees of Freedom? • Problem: • ____________________________________ • ____________________________________ • Can use simpler, more conservative method. • ____________________________________

  15. Inference for μ1 - μ2 • C% confidence interval for μ1 - μ2 • t* is critical value from t distribution table. • d.f. = n1 – 1 or n2 – 1, whichever is smaller.

  16. Example #1 • A statistics student designed an experiment to test the battery life of two brands of batteries. For the sample of 6 generic batteries, the mean amount of time the batteries lasted was 206.0 minutes with a standard deviation of 10.3 minutes. For the sample of 6 name brand batteries, the mean amount of time the batteries lasted was 187.4 minutes with a standard deviation of 14.6 minutes. Calculate a 90% confidence interval for the difference in battery life between the generic and name brand batteries.

  17. Example #1 (cont.) • Assumptions: • Random samples • OK • Independent samples • different batteries for each sample. • Nearly Normal • data shows no real outliers.

  18. Example #1 (cont.) • d.f. = 5 • μ1 = ______________________________ • μ2 = ______________________________

  19. Example #1 (cont.)

  20. Example #1 (cont.)

  21. Example #2 • The Core Plus Mathematics Project was designed to help students improve their mathematical reasoning skills. At the end of 3 years, students in both the CPMP program and students in a traditional math program took an algebra test (without calculators). The 312 CPMP students had a mean score of 29.0 and a standard deviation of 18.8 while the 265 traditional students had a mean score of 38.4 with a standard deviation of 16.2. Calculate a 95% confidence interval for the mean difference in scores between the two groups.

  22. Example #2 (cont.) • Assumptions: • Random samples • no reason to think non-random • Independent samples • different students in each group • Nearly Normal • n1 and n2 are large, so not important.

  23. Example #2 (cont.) • d.f. = smaller of 311 and 264 = 264 • μ1 = _____________________________ • μ2 = _____________________________

  24. Example #2 (cont.)

  25. Example #2 (cont.)

  26. Hypothesis Test for μ1 - μ2 • HO: __________________________ • HA: Three possibilities • HA: ______________________________ • HA: ______________________________ • HA: ______________________________

  27. Hypothesis Test for μ1 - μ2 • Assumptions • Random samples. • Independent samples. • Nearly Normal Population Distributions.

  28. Hypothesis Test for μ1 - μ2 • Test statistic: • d.f. = smaller of n1 – 1 and n2 – 1

  29. P-value for Ha:__________________ • P-value = P(t d.f. > t)

  30. P-value for Ha: _________________ • P-value = P(t d.f. < t)

  31. P-value for Ha:__________________ • P-value = 2*P(t d.f. > |t|)

  32. Hypothesis Test for μ1 - μ2 • Small p-value • _____________________________________ _____________________________________ • Large p-value • _____________________________________ • _____________________________________

  33. Decision • If p-value < α __________________________________ __________________________________ • If p-value > α __________________________________ __________________________________

  34. Hypothesis Test for μ1 - μ2 • Conclusion: Statement about value of μ1 - μ2. Always stated in terms of problem.

  35. Example #1 • Back to the reading example. The educator takes a random sample of all third graders in a large school district and divides them into the two groups. The mean score of the 18 students in the new activities group was 51.72 with a standard deviation of 11.71. The mean score of the 20 students in the traditional group was 41.8 with a standard deviation of 17.45. Is this evidence that the students in the new activities group have a higher mean reading score? Use α = 0.1.

  36. Example #1 (cont.) • μ1 = ______________________________ • μ2 = ______________________________

  37. Example #1 (cont.) • HO: ____________ • HA: ____________ • Assumptions: • Random Samples • OK • Independent Samples • Different set of students in each group. • Nearly Normal • boxplots look symmetric

  38. Example #1 (cont.)

  39. Example #1 (cont.) • d.f. = smaller of 17 or 19 = 17 • P-value

  40. Example #1 (cont.) • Decision:

  41. Example #1 (cont.) • Conclusion:

  42. Example #2 • In June 2002, the Journal of Applied Psychology reported on a study that examined whether the content of TV shows influenced the ability of viewers to recall brand names of items featured in commercials. Researchers randomly assigned volunteers to watch either a program with violent content or a program with neutral content. Both programs featured the same 9 commercials. After the shows ended, subjects were asked to recall the brands in the commercials. Is there evidence that viewer memory for ads differs between programs? Use α = 0.05

  43. Example #2 (cont.) • μ1 = _____________________________ • μ2 = _____________________________

  44. Example #2 (cont.) • HO: ______________ • HA: ______________ • Assumptions: • Random Samples • no reason to think not random • Independent Samples • Different people in each group. • Nearly Normal • n1 and n2 are large so not important

  45. Example #2 (cont.)

  46. Example #2 (cont.) • d.f. = smaller of 107 or 107 = 107 • P-value

  47. Example #2 (cont.) • Decision:

  48. Example #2 (cont.) • Conclusion:

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