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Estimates and Sample Sizes

Estimates and Sample Sizes. But, first. Example 1:. In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adult population that believe in global warming?

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Estimates and Sample Sizes

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  1. Estimates and Sample Sizes

  2. But, first

  3. Example 1: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adult population that believe in global warming? Notation: p is the population proportion (an unknown parameter). is the sample proportion (computed). From the poll data = 0.70. Apparently, 0.70 will be the best estimate of the proportion of all adults who believe in global warming. pg 329

  4. Definition A point estimate is a single value (or point) used to approximate a population parameter. pg 329

  5. Definition The sample proportion p is the best point estimate of the population proportion p. ˆ

  6. Example (continued) We say that 0.70, or 70% is the best point estimate of the proportion of all adults who believe in global warming. But how reliable (accurate) is this estimate? We will see that its margin of error is 2.3%. This means the true proportion of adults who believe in global warming is between 67.7% and 72.3%. This gives an interval (from 67.7% to 72.3%) containing the true (but unknown) value of the population proportion.

  7. What Is the Problem? We have a point estimate, but we don’t know how good it is. We need to have a range of values.

  8. Definition A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. pg 329

  9. Notation for Critical Value α= Greek letter alpha Usually the confidence interval is 1- α

  10. Definition A confidence level is the probability 1 –  (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter. The confidence level is also called degree of confidence, or the confidence coefficient. Most common choices are 90%, 95%, or 99%. ( = 10%), ( = 5%), ( = 1%) pg 330

  11. Example (continued) In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. The sample proportion = 0.70 is the best estimate of the population proportion p. A 95% confidence interval for the unknown population parameter is 0.677 < p < 0.723 What does it mean, exactly?

  12. Interpreting a Confidence IntervalCorrect We are 95% confident that the interval from 0.677 to 0.723 actually does contain the true value of the population proportion p. This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals, then 95% of them would actually contain the value of the population proportion p. pg 330

  13. Interpreting a Confidence IntervalIncorrect There is a 95% chance that the true value of p will fall between .0677 and .0723. It would also be incorrect to say that 95% of sample proportions fall between .0677 and .0723 pg 330

  14. Caution Know the correct interpretation of a confidence interval. It is wrong to say “the probability that the population parameter belongs to the confidence interval is 95%” because the population parameter is not a random variable, it does not change its value.

  15. Caution A confidence level of 95% tells us that the process we are using will result in confidence level limits that contain the true population proportion 95% of the time. We expect that 19 out of 20 samples should contain confidence levels that contain the true value of p

  16. Caution Do not confuse two percentages: the proportion may be represented by percents (like 70% in the example), and the confidence level may be represented by percents (like 95% in the example). Proportion may be any number from 0% to 100%. Confidence level is usually 90% or 95% or 99%.

  17. ˆ ˆ p p – E < p < + E p + E (p – E, p + E) ˆ ˆ Confidence Interval for a Population Proportion p ˆ

  18. Finding the Point Estimate and E from a Confidence Interval Point estimate of p: p =(upper confidence limit) + (lower confidence limit) 2 ˆ ˆ Margin of Error: E = (upper confidence limit) — (lower confidence limit) 2

  19. Next we learn how to construct confidence intervals pg 331

  20. Critical Values A z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. The standard normal distribution is divided into three regions: middle part has area 1-α and two tails (left and right) have area α/2 each: pg 331

  21. Critical Values The z scores separate the middle interval (likely values) from the tails (unlikely values). They are zα/2 and –zα/2 , found from Table A-2.

  22. Definition A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.

  23. Notation for Critical Value The critical valuez/2 separates an area of /2in the right tail of the standard normal distribution. The value of –z/2 separates an area of /2 in the left tail. The subscript /2is simply a reminder that the z score separates an area of /2in the tail.

  24. Finding zα/2 Confidence α zα/2 Level 90% 0.10 1.645 95% 0.05 1.96 99% 0.01 2.575 pg 332

  25. zα/2 - zα/2 Critical Values Finding zα/2 for a 95% Confidence Level α = 5% α/2 = 2.5% = .025 pg 332

  26. Definition Margin of error, denoted by E, is the maximum likely difference (with probability 1 – , such as 0.95) between the observed proportion and the true value of the population proportion p. The margin of error E is also called the maximum error of the estimate. pg 333

  27. Margin of Error for Proportions Formula 7-1, pg 333

  28. Notation E = margin of error ^ p = sample proportion ^ ^ q = 1 – p n = number of sample values

  29. ˆ p ˆ p p – E < < + E Confidence Interval for a Population Proportion p where

  30. ˆ ˆ p p p – E < < + E p + E (p – E, p + E) ˆ ˆ Confidence Interval for a Population Proportion p Also written as: ˆ Also written as:

  31. Finding the Point Estimate and E from a Confidence Interval Point estimate of p: p =(upper confidence limit) + (lower confidence limit) 2 ˆ ˆ Margin of Error: E = (upper confidence limit) — (lower confidence limit) 2

  32. Round-Off Rule for Confidence Interval Estimates of p • Round the confidence interval limits for p to three significant digits.

  33. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. Find the margin of error E that corresponds to a 95% confidence level. Find the 95% confidence interval estimate of the population proportion p. Based on the results, can we safely conclude that the majority of adults believe in global warming? Assuming that you are a newspaper reporter, write a brief statement that describes the results accurately, with all relevant information. pg 334

  34. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. Find the margin of error E that corresponds to a 95% confidence level. Zα/2 = 1.96 (Slide 24) p=.70 q=.30 n=1501 E = 1.96 * sqrt(.70*.30/1501) = .023183 Slide 27 pg 334

  35. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. b. Find the 95% confidence interval estimate of the population proportion p. ˆ ˆ p – E < p < p + E .70 - .023183 < p < .70 + .023183 .677 < p < .723 pg 334

  36. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. • Based on the results, can we safely conclude that the majority of adults believe in global warming? • Based on the confidence interval obtained in b. it appears that the proportion of adults who believe in global warming is greater than .5, so we can conclude that the majority of adults believe in global warming. pg 334

  37. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. • Based on the results, can we safely conclude that the majority of adults believe in global warming? • Yes. pg 334

  38. Example 3: In a recent poll, 70% of 1501 randomly selected adults said they believed in global warming. • Assuming that you are a newspaper reporter, write a brief statement that describes the results accurately, with all relevant information. • 70% of U.S. adults believe that the earth is getting warmer, based on a Pew Research Center poll of 1501 randomly selected adults in the US. In theory, in 95% of such polls, the percentage should differ by no more than 2.3% in either direction from the percentage that would be found by interviewing all adults in the U.S. pg 334

  39. Confidence Intervals by TI-83/84 • Press STAT and select TESTS • Scroll down to 1-PropZInt • and press ENTER • Type in x:(number of successes) • n:(number of trials) • C-Level:(confidence level) • Press on Calculate • Read the confidence interval (…..,..…) • and the point estimate p=… ^

  40. Confidence Intervals by Excel Example 3, pg 334. Enter data

  41. Confidence Intervals by Excel Enter formula

  42. Confidence Intervals by Excel

  43. Sample Size pg 336

  44. Sample Size Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

  45. (solve for n by algebra) Determining Sample Size

  46. ˆ When an estimate of p is known: ˆ When no estimate of p is known: Sample Size for Estimating Proportion p

  47. Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of nup to the next larger whole number. Examples: n=310.67 round up to 311 n=310.23 round up to 311 n=310.01 round up to 311 pg 336

  48. Example: A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points? a) In 2006, 73% of adults used the Internet. b) No known possible value of the proportion. Example 4, pg 336

  49. Example: a) Use To be 95% confident that our sample percentage is within three percentage points of the true percentage for all adults, we should obtain a random sample of 842 adults.

  50. Example: b) Use This time we don’t know the sample proportion To be 95% confident that our sample percentage is within three percentage points of the true percentage for all adults, we should obtain a random sample of 1068 adults.

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