Introductory Statistics, Lecture 14

Activity 8.1b „ 1. Your instructor will give you Math 140 Introductory Statistics one of the population proportions whose line segments are missing in Display 8.1. „ 2. Compute the reasonably likely outcomes for your population proportion p. „ 3. On your copy of the chart in Display 8.1, draw a horizontal line segment showing the reasonably likely outcomes for your group’s proportion p. „ 4. Get the reasonably likely outcomes from the other groups in your class and complete the chart with the line segments from those values of p. 1 Activity 8.2 Results p Left 0.25 0.1158 0.3841 0.3 0.1579 0.4420 0.35 0.2021 0.4978 0.4 0.2481 0.5518 0.45 0.2958 0.6041 0.5 0.3450 0.6549 0.55 0.3958 0.7041 0.6 0.4481 0.7518 0.65 0.5021 0.7978 0.7 0.5579 0.8420 0.75 0.6158 0.8841 2 Example: Plausible Percentages „ In a group of 40 adults, exactly 30 were right-eye dominant. Right Assuming this can be considered a random sample of all adults, is it plausible that if you tested all adults, you would find that 50% are right-eye dominant? Is it plausible that 80% are righteye dominant? What percentages are plausible? 3 4 Confidence Interval Discussion: Confidence Intervals „ D3. According to the 2000 U.S. Census, about 60% of Hispanics „ A 95% confidence interval consists of those in the United States are of Mexican origin. Would it be reasonably likely in a survey of 40 randomly chosen Hispanics to find that 27 are of Mexican origin? population percentages p for which the sample proportion p.̂ is reasonably likely. Source: www.census.gov/prod/2001pubs/c2kbr01-3.pdf. „ D4. According to the 2000 U.S. Census, about 30% of people „ Notes: over age 85 are men. In a random sample of 40 people over age 85, would it be reasonably likely to get 60% who are men? In the previous diagram the horizontal segments were reasonably likely intervals. „ The 95% confidence intervals should be represented as vertical segments. In this case p is the unknown parameter. In the previous example the 95% confidence interval goes from 0.6 to about 0.85 „ Source: www.census.gov/prod/2001pubs/c2kbr01-10.pdf. „ D5. Suppose that in a random sample of 40 toddlers, 34 know what color Elmo is. What is the 95% confidence interval for the percentage of toddlers who know what color Elmo is? „ D6. Polls usually report a margin of error. Suppose a poll of 40 randomly selected statistics majors finds that 20 are female. The poll reports that 50% of statistics majors are female, with a margin of error of 15%. Use your completed chart to explain where the 15% came from. 5 6 A Confidence Interval for a Proportion. (Any percent) Getting a Formula „ A confidence interval for the proportion of successes p in the „ The endpoints of the horizontal population is given by the formula segment (a reasonably likely interval) are: p ± 1.96 p (1 − p ) n pˆ ± z ⋅ „ The horizontal segments are „ n is the sample size, p̂ is the proportion of successes in the sample. delimited by two curves that are almost parallel lines of slope 1. „ Thus the vertical segment has about the same length as the horizontal segment. „ Moreover, since the center of each horizontal segment is p, then the endpoints of the vertical segment are: pˆ ± 1.96 pˆ (1 − pˆ ) n The value of z depends on how confident you want to be that p̂ will be in the confidence interval. For a 95% confidence interval, use z = 1.96; for a 90% confidence interval, use z = 1.645; for a 99% confidence interval, use z = 2.576; and so on. „ This confidence interval is reasonably accurate when three conditions are met: „ The sample was a simple random sample from a binomial population (every subject is either a success or a failure). ˆ and n(1 − pˆ ) are at least 10. „ Both np „ The size of the population is at least 10 times the size of the sample n. pˆ (1 − pˆ ) n 7 8 A Confidence Interval and the Margin of Error „ A confidence interval for the proportion of successes p in the population is given by the formula pˆ ± z ⋅ „ This confidence interval is „ „ 1.645 for 90% confidence  z =  1.96 for 95% confidence 2.576 for 99% confidence  E =z⋅ „ At the beginning of this section, you read reasonably accurate when three conditions are met: pˆ (1 − pˆ ) n „ The quantity Example: Unattached singles „ about a recent Pew Research Center survey that reported that a 55% of singles ages 1829 say that they unattached. The survey reported a margin of error of 3%. Use the formula for a 95% confidence interval for p to verify this margin of error. Then describe specifically what p stands for in this situation. The sample was a SRS from a binomial population (every subject is either a success or a failure). npˆ ≥ 10 and n(1 − pˆ ) ≥ 10. The size of the population is at least 10 times the size of the sample n. pˆ (1 − pˆ ) n is called the margin of error. It is half the length of the confidence interval. 9 Activity 8.3 (Simulated) Activity 8.3 (Simulated) „ Examples: 1. Generate a random sample of 2. 3. 4. 5. 6. 40 numbers between 0 and 9. Count the number of even digits in your sample of 40. Construct a 95% confidence interval for the proportion of random digits that are even. Repeat a 100 times and draw all of the intervals in the appropriate display (like Display 8.5 in p. 426) What is the true proportion of all random digits that are even? What percentage of the confidence intervals captured the true proportion? Is this what you expected? Explain. 10 „ Examples: 16963091312835600 06783149656967532 681412 20 pˆ = = 0.5 Even: 20 40 95% Confidence Interval 16963091312835600 06783149656967532 681412 20 pˆ = = 0.5 Even: 20 40 95% Confidence Interval 0.5(1 − 0.5) = 40 (0.34505, 0.65495) „ 89442668749634882 74420361605209827 217258 27 pˆ = = 0.675 Even: 27 40 95% Confidence Interval 0.5(1 − 0.5) = 40 (0.34505, 0.65495) „ 89442668749634882 74420361605209827 217258 27 pˆ = = 0.675 Even: 27 40 95% Confidence Interval 0.5 ± (1.96) 0.5 ± (1.96) 0.675(1 − 0.675) = 40 (0.52985, 0.82015) 0.675(1 − 0.675) = 40 (0.52985, 0.82015) 0.675 ± (1.96) 0.675 ± (1.96) 11 12 Activity 8.3 (Simulated) Back to Opinion Polls. „ At the beginning of this section, you read about a recent Pew Research Center survey that reported that a 55% of singles ages 18-29 say that they aren’t in a committed relationship and are not actively looking for a romantic partner. (We’ll call these young singles “unattached”). The survey reported a margin of error of 3%. You now should be able to answer these questions: „ What is it that you are 95% sure is in the confidence How many intervals contain the true proportion of even numbers p = 0.5? Answer: all except one, that is 38 out of 39. Or 97.43% of all of the intervals. interval? Answer: The proportion of all Americans 18-29 years old that consider themselves unattached. In general, if we randomly calculate a large number of 95% Confidence Intervals we should expect that about 95% of them will contain the true value of p. 13 14 Back to Opinion Polls. Back to Opinion Polls. „ At the beginning of this section, you read about a „ At the beginning of this section, you read about a recent Pew Research Center survey that reported that a 55% of singles ages 18-29 say that they aren’t in a committed relationship and are not actively looking for a romantic partner. (We’ll call these young singles “unattached”). The survey reported a margin of error of 3%. You now should be able to answer these questions: recent Pew Research Center survey that reported that a 55% of singles ages 18-29 say that they aren’t in a committed relationship and are not actively looking for a romantic partner. (We’ll call these young singles “unattached”). The survey reported a margin of error of 3%. You now should be able to answer these questions: „ What is the meaning of “95% confidence”? „ What is the interpretation of the confidence interval of Answer: If we were to take 100 random samples of Americans 18-29 and compute the 95% confidence interval from each sample, then we expect that 95 of them will contain the true proportion of all Americans 18-29 that consider themselves unattached. (whatever that proportion is). 52% to 58%? Answer: We are 95% confident that if we could ask all Americans 18-29, then between 52% and 58% would consider themselves unattached. 15 16 What sample size should you use? What sample size should you use? „ Example. (p.426) Suppose you take a survey „ Simple Answer: In general, the larger the sample the more accurate the results will be (smaller margin of error). If we would like to find the sample size for a particular margin of error, all we have to do is solve for n. and get p̂ = 0.7. If your sample size is 100, what would be the margin of error for a 95% confidence interval? E =z⋅ pˆ (1 − pˆ ) 0.7(1 − 0.7) = (1.96) = .0898 n 100 „ If you quadruple your sample, what would be the new margin of error? E =z⋅ pˆ (1 − pˆ ) n  pˆ (1 − pˆ )  E = z ⋅   n  2 pˆ (1 − pˆ ) 0.7(1 − 0.7) E =z⋅ = (1.96) = .0449 n 400 n= 17 Example: Estimating needed sample sizes (p.481) „ What sample size should you use for a survey if you want the margin of error to be at most 3% with 95% confidence but you have no estimate of p? „ D17. Suppose it costs $1 to survey each person in your sample. You judge that p is about 0.5. What will your survey cost if you want a margin of error of about 10%? 1%? 0.1%? 19 2 If you do not have an estimate for then you should use 0.5. p̂ , You may get a larger n than you need but never a smaller one. z 2 ⋅ pˆ (1 − pˆ ) E2 18