8.1 Estimating a Proportion with Confidence � A recent Phi Delta Kappa/Gallup poll reported that a record 51% Math 140 of the American public assigns a grade of A or B to the public schools in their community and that this survey had a margin of Introductory Statistics error of 3%. Source: 2001, www.gallup.com/poll/releases/pr010823.asp. � These results are based on telephone interviews with a randomly selected national sample of 1108 adults, 18 years and older, conducted May 23–June 6, 2001. Professor Silvia Fernández � For results based on this sample, one can say with 95 percent confidence that the maximum error attributable to sampling and Chapter 8 other random effects is plus or minus 3 percentage points. In Based on the book Statistics in Action addition to sampling error, question wording and practical by A. Watkins, R. Scheaffer, and G. Cobb. difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls. 8.1 Estimating a Proportion Reasonably Likely (Again) with Confidence � A recent Phi Delta Kappa/Gallup poll reported that a record 51% � We learned in 7.3 that if we get a sample of size n of the American public assigns a grade of A or B to the public from a population with proportion of success p , then schools in their community and that this survey had a margin of the reasonably likely outcomes fall between the error of 3%. Source: 2001, www.gallup.com/poll/releases/pr010823.asp. values � The Gallup organization is disclosing that they didn’t ask all − p p ( 1 ) adults in the United States, only 1108. Even so, unless there are ± p 1 . 96 some special difficulties such as problems with the wording of n the question, they are 95% confident that the error is less than 3% either way in the percentages they report. That is, they are 95% confident that if they were to ask all adults in the United � Recall that reasonably likely outcomes are those in States to give a grade to the public schools, 51% ± 3%, or the middle 95% of the distribution of all possible between 48% and 54%, would give a grade of A or B. How can the Gallup organization possibly make such a statement? outcomes. The outcomes in the upper 2.5% and the lower 2.5% of the distribution are rare events—they happen, but rarely. 1
Examples Activity 8.1b 1. Your instructor will give you � � Example p.468. Suppose that you will flip a fair coin one of the population propor- 100 times. What are the reasonably likely values of tions whose line segments are missing in Display 8.1. the sample proportion ? What numbers of heads p ˆ are reasonably likely? 2. Compute the reasonably � likely outcomes for your population proportion p . � D1. Suppose 35% of a population think they pay too much for car insurance. A polling organization takes 3. On your copy of the chart in � Display 8.1, draw a horizontal a random sample of 500 people in this population line segment showing the and computes the sample proportion of people who reasonably likely outcomes for your group’s proportion p . think that they pay too much for car insurance. � a. There is a 95% chance that will be between what p ˆ 4. Get the reasonably likely � two values? outcomes from the other groups in your class and � b. Is it reasonably likely to get 145 people in the complete the chart with the line sample who think they pay too much for car insurance? segments from those values of p . Activity 8.2 Results Example: Plausible Percentages � In a group of 40 adults, exactly 30 were right-eye dominant. p Left Right Assuming this can be considered a random sample of all adults, 0.25 0.1158 0.3841 is it plausible that if you tested all adults, you would find that 0.3 0.1579 0.4420 50% are right-eye dominant? Is it plausible that 80% are right- 0.35 0.2021 0.4978 eye dominant? What percentages are plausible? 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
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 population percentages p for which the to find that 27 are of Mexican origin? . sample proportion p is reasonably likely. ˆ 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 Source: www.census.gov/prod/2001pubs/c2kbr01 - 1 0.pdf. � D5. Suppose that in a random sample of 40 toddlers, 34 know segments were reasonably likely intervals. what color Elmo is. What is the 95% confidence interval for the percentage of toddlers who know what color Elmo is? � The 95% confidence intervals should be � D6. Polls usually report a margin of error. Suppose a poll of 40 represented as vertical segments. In this case randomly selected statistics majors finds that 20 are female. The p is the unknown parameter. In the previous poll reports that 50% of statistics majors are female, with a margin of error of 15%. Use your completed chart to explain example the 95% confidence interval goes where the 15% came from. from 0.6 to about 0.85 A Confidence Interval Getting a Formula for a Proportion. (Any percent) � 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 ˆ ( 1 p ˆ ) − ± ⋅ p ( 1 p ) ± p ˆ z p 1 . 96 n n The horizontal segments are � � n is the sample size, is the proportion of successes in the sample. delimited by two curves that are ˆ p almost parallel lines of slope 1. The value of z depends on how confident you want to be that will ˆ p be in the confidence interval. For a 95% confidence interval, use z � Thus the vertical segment has = 1.96; for a 90% confidence interval, use z = 1.645; for a 99% about the same length as the confidence interval, use z = 2.576; and so on. horizontal segment. Moreover, since the center of � � This confidence interval is reasonably accurate when three each horizontal segment is p , conditions are met: then the endpoints of the � The sample was a simple random sample from a binomial vertical segment are: population (every subject is either a success or a failure). − − � Both and ( ) are at least 10. n p ˆ n 1 ˆ p ˆ p ( 1 ˆ p ) ± p ˆ 1 . 96 � The size of the population is at least 10 times the size of the n sample n . 3
A Confidence Interval Example: Safety Violations. and the Margin of Error � A confidence interval for the � This confidence interval is � Suppose you have a random sample of 40 proportion of successes p in the reasonably accurate when three population is given by the conditions are met: buses from a large city and find that 24 have formula a safety violation. Find the 90% confidence − � The sample was a SRS ˆ p ( 1 p ˆ ) ± ⋅ p ˆ z from a binomial population interval for the proportion of all buses that n (every subject is either a success or a failure). have a safety violation. ≥ and ( − ) ≥ n p ˆ 10 n 1 p ˆ 10 . � ⎧ for 90% confidence 1 . 645 � The size of the population is ⎪ at least 10 times the size of = ⎨ for 95% confidence z 1 . 96 the sample n . ⎪ for 99% confidence ⎩ 2 . 576 The quantity � − p ˆ ( 1 p ˆ ) = ⋅ E z n is called the margin of error . It is half the length of the confidence interval. Activity 8.3 (Simulated) Activity 8.3 (Simulated) Examples: Examples: � � 1. Generate a random sample of 1 6 9 6 3 0 9 1 3 1 2 8 3 5 6 0 0 1 6 9 6 3 0 9 1 3 1 2 8 3 5 6 0 0 40 numbers between 0 and 9. 0 6 7 8 3 1 4 9 6 5 6 9 6 7 5 3 2 0 6 7 8 3 1 4 9 6 5 6 9 6 7 5 3 2 6 8 1 4 1 2 6 8 1 4 1 2 2. Count the number of even digits 20 20 = = = = in your sample of 40. Even: 20 Even: 20 ˆ p 0 . 5 p ˆ 0 . 5 40 40 3. Construct a 95% confidence 95% Confidence Interval 95% Confidence Interval interval for the proportion of random digits that are even. − − 0 . 5 ( 1 0 . 5 ) 0 . 5 ( 1 0 . 5 ) ± = ± = . . 0 5 ( 1 . 96 ) 0 5 ( 1 . 96 ) 4. Repeat a 100 times and draw all 40 40 of the intervals in the appropriate display (like Display ( 0 . 34505 , 0 . 65495 ) ( 0 . 34505 , 0 . 65495 ) 8.5 in p. 426) 8 9 4 4 2 6 6 8 7 4 9 6 3 4 8 8 2 8 9 4 4 2 6 6 8 7 4 9 6 3 4 8 8 2 � � 7 4 4 2 0 3 6 1 6 0 5 2 0 9 8 2 7 7 4 4 2 0 3 6 1 6 0 5 2 0 9 8 2 7 5. What is the true proportion of all 2 1 7 2 5 8 2 1 7 2 5 8 random digits that are even? 27 27 = = = = ˆ p 0 . 675 p ˆ 0 . 675 Even: 27 Even: 27 6. What percentage of the 40 40 confidence intervals captured 95% Confidence Interval 95% Confidence Interval the true proportion? Is this what − − you expected? Explain. 0 . 675 ( 1 0 . 675 ) 0 . 675 ( 1 0 . 675 ) ± = ± = . . 0 675 ( 1 . 96 ) 0 675 ( 1 . 96 ) 40 40 ( 0 . 52985 , 0 . 82015 ) ( 0 . 52985 , 0 . 82015 ) 4
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