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Confidence Interval for Sample Proportions: Estimating Enrollment in School

This chart provides a complete guide for estimating the proportion of enrolled students in a sample of 40 randomly chosen 20- and 21-year-olds, based on a recent report stating that about 30% are enrolled in school. It also includes information on calculating confidence intervals for proportions and how sample size affects the margin of error.

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Confidence Interval for Sample Proportions: Estimating Enrollment in School

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  1. Section 8.1 Day 2

  2. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

  3. According to a recent report, about 30% of 20- and 21-year-olds are enrolled in school. How many enrolled students is it reasonably likely to get in a sample of 40 randomly chosen 20- and 21-year-olds?

  4. According to a recent report, about 30% of 20- and 21-year-olds are enrolled in school. How many enrolled students is it reasonably likely to get in a sample of 40 randomly chosen 20- and 21-year-olds? Populationsample

  5. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

  6. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

  7. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472) 6 to 18 students

  8. Suppose that in a random sample of 40 adults, 26 of the adults have never married. Find the 95% confidence interval for the proportion of all adults who have never married.

  9. Suppose that in a random sample of 40 adults, 26 of the adults have never married. Find the 95% confidence interval for the proportion of all adults who have never married. Samplepopulation

  10. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

  11. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472)

  12. A Complete Chart of Reasonably Likely Sample Proportions for n = 40 (Page 472) (0.5, 0.8)

  13. A confidence interval (CI) for the proportion of successes p in the population is given by the formula Here n is the sample size, p is the proportion of successes in the sample, and z* is the critical value.

  14. Value of z* depends on how confident you want to be that p will be in the confidence interval.

  15. Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use z* =1.645

  16. Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use z* = 1.645 For 95% confidence interval, use z* = 1.96

  17. Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use z* = 1.645 For 95% confidence interval, use z* = 1.96 For 99% confidence interval, use z* = 2.576

  18. As we become more confident, what happens to the width of the CI? For 90% confidence interval, use z* = 1.645 For 95% confidence interval, use z* = 1.96 For 99% confidence interval, use z* = 2.576

  19. Check Conditions This confidence interval is reasonably accurate when three conditions are met:

  20. Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population

  21. Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population (2) Both np and n(1 – p) are at least 10

  22. Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population (2) Both np and n(1 – p) are at least 10 (3) Size of the population is at least 10 times the size of the sample

  23. A confidence interval for the proportion of successes p in the population is given by the formula

  24. A confidence interval for the proportion of successes p in the population is given by the formula

  25. A confidence interval for the proportion of successes p in the population is given by the formula Standard error

  26. The quantity is called the margin of error.

  27. The quantity is called the margin of error.

  28. Why is it true that when everything else is held constant, the margin of error is smaller with a larger sample size than with a smaller sample size?

  29. Suppose you take a random sample and get p = 0.7. a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error.

  30. Suppose you take a random sample and get p = 0.7. a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error.

  31. Suppose you take a random sample and get p = 0.7. a) If your sample size is 100, find the 95% confidence interval for p, and state the margin of error. ≈ 0.7 ± 0.0898 So, the 95% CI is (0.6101, 0.7898) and the margin of error is approximately 0.0898

  32. Suppose you take a random sample and get p = 0.7. b) What happens to the confidence interval and margin of error if you quadruple the sample size to 400?

  33. Suppose you take a random sample and get p = 0.7. b) What happens to the confidence interval and margin of error if you quadruple the sample size to 400? The 95% CI is (0.6551, 0.7449) and the margin of error is approx. 0.0449.

  34. For n = 100: The 95% CI is (0.6101, 0.7898) and the margin of error is approximately 0.0898 For n = 400: The 95% CI is (0.6551, 0.7449) and the margin of error is approx. 0.0449. Thus, quadrupling the sample size cuts the margin of error in half.

  35. How does a confidence interval for large sample sizes compare to those for small sample sizes?

  36. How does a confidence interval for large sample sizes compare to those for small sample sizes? CI’s for large sample sizes are narrower than those for small sample sizes. Why should this make sense to us?

  37. CI’s for large sample sizes are narrower than those for small sample sizes. This makes sense as the larger the sample size, the closer p should be to p.

  38. What sample size should you use?

  39. What sample size should you use? The larger the sample size, the more precise the results will be.

  40. What sample size should you use? The larger the sample size, the more precise the results will be. If larger is “better”, why limit our sample size?

  41. Researchers always have limited time and money. For practical reasons, we have to limit the size of our samples.

  42. Researchers always have limited time and money. For practical reasons, we have to limit the size of our samples. So, how large is sufficient?

  43. How large is sufficient depends on what margin of error is acceptable to you. Solve this formula for n. n = ?

  44. To use this formula, you need three pieces of information.

  45. To use this formula, you need three pieces of information: (1) you have to know the margin of error that is acceptable

  46. To use this formula, you need three pieces of information: (1) you have to know the margin of error that is acceptable (2) you have to decide on a level of confidence so you know what value of z* to use

  47. To use this formula, you need three pieces of information: (1) you have to know the margin of error that is acceptable (2) you have to decide on a level of confidence so you know what value of z* to use (3) you have to have an estimate of p.

  48. To use this formula, you need three pieces of information: (1) you have to know the margin of error that is acceptable (2) you have to decide on a level of confidence so you know what value of z* to use (3) you have to have an estimate of p. --if you have a good estimate of p, use it in this formula

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