ࡱ> uwt'` 0bjbj{P{P 4::Pnnnnnnn 6V6V6V8nV2W.RX@Y@Y@YXY7Z7Z7Z$whtin7Z3Z7Z7Z7Znn@YXY?\\\7Zn@YnXY\7Z\\Znn@YFX a 6VO[4`,U0BTS[SSnx 7Z7Z\7Z7Z7Z7Z7ZG\7Z7Z7Z7Z7Z7Z7Z...!:D...:T@ nnnnnn Use of margin of error in the media Introduction For this activity, an article is used as source material for a sequence of questions. Questions can then be written that assess students skills at evaluating polls, including their understanding of margin of error, based on claims made within the article. The questions written should be based on claims made in the article, requiring a mixture of knowledge and interpretation to answer. This is an approach that can be applied to other similar articles (designing specific questions for an article). Prior to working with margin of errors and confidence intervals for proportions, students should have experienced sampling variability with categorical data (sample proportions/percentages) involving different sample sizes (see movies on  HYPERLINK "http://www.censusatschool.org.nz/2009/informal-inference/WPRH/" http://www.censusatschool.org.nz/2009/informal-inference/WPRH/). Questions should elicit responses that show an understanding of the following (see  HYPERLINK "http://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdf" http://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdf for more information): Poll percentages are used to make inferences about the population and the margin of error is needed to account for sampling variability Who is the population that the inferences are being made for The margin of error reported in the media is based on p = 0.5 and 95% confidence level The margin of error is roughly the same as  EMBED Equation.3 , where n is the size of the sample why? The reported margin of error is pretty good for poll percentages (sample proportions) from 30% to 70%, but too large for other values why? The reported margin of error can not be used for poll percentages for sub-groups of the population sampled why? The margin of error can be estimated using general knowledge about the population e.g. gender The reported margin of error can not be used to compare two poll percentages from two independent samples, or to compare two poll percentages within the same sample The relationships between the margin of error, poll percentage and sample size The margin of error measures the sampling error (or variability) but there are other associated errors i.e. non-sampling errors such as non-response error, etc. The confidence interval gives a range of plausible values for the true population percentage with 95% confidence A 95% confidence interval means that in the long run for only approximately 19 out of every 20 (95%) surveys taken, the confidence interval will contain the true population percentage The structure of a confidence interval ( estimate margin of error Beyond the scope of level eight (but you can read more about in this document  HYPERLINK "http://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdf" http://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdf ): Comparing two poll percentages from two independent samples Comparing two poll percentages within the same sample The article used for this activity is found here Pollsters NZ election winners too  HYPERLINK "http://www.nzherald.co.nz/nz-election-2008/news/article.cfm?c_id=1501799&objectid=10542038" http://www.nzherald.co.nz/nz-election-2008/news/article.cfm?c_id=1501799&objectid=10542038 All the final results for the 2008 election are available from  HYPERLINK "http://www.electionresults.govt.nz" www.electionresults.govt.nz Statistically based reportGet students to read the article Pollsters NZ election winners too published in the New Zealand Herald (Sunday 9th November 2008) and available online at  HYPERLINK "http://www.nzherald.co.nz" www.nzherald.co.nz. All of the final results for the 2008 election are available from  HYPERLINK "http://www.electionresults.govt.nz" www.electionresults.govt.nz. This information presented can be summarised into this table: Actual election results Final Herald Digi-poll National Party 45.45 47.9 Labour Party 33.77 36.4 Green Party 6.43 5.8 New Zealand First Party 4.21 3.9 Mori Party 2.24 2.3 ACT New Zealand 3.72 1.8 United Future 0.89 0.7 Jim Anderton's Progressive 0.93 0.2 Number sampled 2,103,842 981 Reported margin of error 3.2% * The poll of 981 respondents was conducted between October 29 and November 5, with a reported margin of error of 3.2 per cent. The undecideds were 6.2 per cent. EvaluationTeacher note: The poll percentages used below are from the Digi-poll results reported in the article. The article says that the polls predicted that National would score in the mid-to-late forties. Explain, with some calculations, whether or not this claim is justified: Construct a confidence interval based on the poll results for the actual percentage of support for the National party, using the margin of error reported in the article. Confidence interval ( estimate margin of error 0.479 0.032 = [0.447, 0.521] Can use margin of error reported because 0.479 is between 0.3 and 0.7 Interpret the confidence interval constructed. Somewhere between 44.7% and 52.1% of New Zealand voters will vote for the National party. We can make this statement with 95% confidence. Verify the claim. Since the article claims that National would receive 45% - 49% of the votes (mid to late forties), the claim is not justified. Based on the confidence interval constructed, the claim should be mid forties to early fifties. Consider how the article made its claim. The article appears to have made its claim based on point estimates from three separate poll results: Herald 47.9%, Roy Morgan 42%, Farifax Nelson 49% The article reports a margin of error of 3.2%. How was the margin of error calculated and what does it measure? Show how the margin of error of 3.2% was calculated. moe H"  EMBED Equation.3 = 0.032 (Teacher note: This is equivalent to  EMBED Equation.3 ) Explain what the margin of error measures. Since the sample was taken from the population, the percentages reported in the poll will not be exact. The margin of error quantifies the sampling variability for a true population percentage of 50%, to give the maximum margin of error. According the article, polling organisations get a lot of flack when they get polls wrong. Discuss the two types of potential sources of errors for polls. Sampling errors. The sampling error is caused by the act of sampling; it has the potential to be bigger in smaller samples; and, provided some form of random sampling has been employed, we can determine how large it can be (the margin of error). Sampling error is unavoidable; it is part of the cost of sampling. We are unable to determine which surveys are the 5% rogue surveys but take comfort in the fact that the 95%-confidence- interval-building process works most (approximately 95%) of the time and, as such, rogue surveys are very few and far between. Non-sampling errors. Polls are often taken in the evening, and so there is a selection bias, as people who work shifts or in the evenings can not take part. If landline phone numbers are used to contact possible respondents (using random digit dialling), this again is a selection bias, as people with only mobile phones or without a landline can not take part. A low response rate for the poll (response bias) maybe a concern because those who response may have substantially different views from those who to not choose to complete the poll. The article says that the polls didnt seem to predict that NZ first party would obtain over 4% of the vote. Explain, with some calculations, whether or not this claim is justified: Construct a confidence interval based on the poll results for the actual percentage of support for the NZ first party, using a margin of error based on 3.9% (the poll result for NZ first). Moe =  EMBED Equation.3 = 0.012 Confidence interval ( estimate margin of error 0.039 0.012 = [0.027, 0.051] Cant use margin of error reported because 0.039 is not between 0.3 and 0.7 Interpret the confidence interval constructed. Somewhere between 2.7% and 5.1% of New Zealand voters will vote for the NZ first party. We can make this statement with 95% confidence. Verify the claim. The claim is not justified because the confidence interval gives a range of plausible values for the true population percentage, which includes values higher than 4% Consider how the article made its claim. It appears the point estimate of 3.9% was used without considering sampling variability. The article talks about the variability in poll results for the Green party, with some as much as 10 percent and others on nine percent. The article then says these disparities largely disappear when the margin of error is taken into account. What do they mean by this statement? (see answer structure above) Another NZ Herald Digi-poll of 750 respondents was conducted between February 27 and March 6 2008. It was found that 35.4% of males and 42.9% of females surveyed said they would vote for the Labour party. Estimate the margin of error for the true percentage of females who will vote for the Labour party. Estimate the number of females in the sample. Assume 50% female, so 750 x 0.5 H" 375 Estimate the margin of error for female sub-group. moe H"  EMBED Equation.3 = 0.052 As 42.9% lies between 30% and 70% The Digi-poll also found that support for the National party was higher in Auckland (55.7%) than the rest of New Zealand (46.7%). Estimate the margin of error the true percentage of Aucklanders who will vote for the National party. Estimate the number of Aucklanders in the sample. Assume 25% from Auckland, so 750 x 0.25 H" 188 Estimate the margin of error for the Auckland sub-group. moe H"  EMBED Equation.3 = 0.073 As 55.7% lies between 30% and 70% ArticlePollsters NZ election winners too NZ Herald 12:22PM Sunday Nov 09, 2008 Aside from John Key, there were other winners last night - the pollsters. New Zealand's election result was close to the final poll results from five organisations that predicted National would score in the mid-to-late forties and Labour in the early-to-mid-thirties. The final Herald Digi-poll put National on 47.9 per cent with Labour on 36.4. Election results from last night saw National score 45 per cent while Labour hit 34 per cent. Other polls were also close to the result. The final poll from Roy Morgan had Labour on 34.5 while National scored 42 per cent. The last Fairfax Nielson poll had National on 49 per cent and Labour on 31. Victoria University of Wellington political science lecturer Stephen Levine said the pollsters were being proven correct last night. Speaking to nzherald.co.nz as the results came in, Dr Levine said the pollsters had generally got it right." Polling organisations get a lot of criticism when they're wrong and there is a lot of variation in the polls but the big picture of all the polls is being reflected on the night so far," Dr Levine said. 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He said one interesting result was New Zealand First polling over four per cent. "I'm not sure the polls were having New Zealand First over four per cent," Dr Levine said. And he was right. 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jhQh:6OJQJ^JhQh:6OJQJ^JhzOJQJ^Jhzhz6OJQJ]^J+jhzhz6EHOJQJU]^J1jO hzOJQJUV_H mH nH sH tH !jhz6OJQJU]^Jhz6OJQJ]^Jh:OJQJ^JhQh:OJQJ^Jh:OJQJ]^JhV/hB 6OJQJ^J))))))))))))))))$*A*p*w*****n+o+p+y+++++++ķķѝķwlaVhkn6OJQJ^Jh:6OJQJ^JhZOJQJ]^JhknOJQJ]^Jh:OJQJ]^Jh:h&6OJQJ]^Jh:OJQJ^Jhh:6OJQJ]^Jh:6OJQJ]^Jh:6OJQJ]^Jh:OJQJ^JhQh:6OJQJ^Jh:6OJQJ^Jhz6OJQJ^J!))))!*"*****o+p+++++.-/---. $IfgdMry $Ifgd\ & F*$Ifgdkn h$If^hgd: h$If^hgd: & F*$Ifgd: $Ifgd:+++,L,,,--.-/-B-S-----..,/v/x////µyyl\yRGhK6OJQJ^JhBOJQJ^JhMryhMry6OJQJ]^Jh 6OJQJ]^JhMryOJQJ^JhOJQJ^Jh OJQJ^Jh>nOJQJ^JhBOJQJ^Jh&ROJQJ^Jh6OJQJ]^Jh\6OJQJ]^Jh\OJQJ^JhQh\OJQJ^Jh:OJQJ]^Jh:h:6OJQJ^J.."/(/x/z///0>0?0'1(1Z1]122 h$If^hgd` & F-$IfgdUt $Ifgd` $IfgdUt $IfgdC h$If^hgdK h$If^hgdB h$If^hgdMry & F+$IfgdMry $IfgdMry//000000>0?0B0C0M0R000&1'1(1J1_1w12̷喋sh^T^G:h`6OJQJ]^JhUt6OJQJ]^JhUtOJQJ^Jh`OJQJ^Jh`OJQJ]^JhQh&ROJQJ^JhBOJQJ]^Jh+~OJQJ]^Jh:OJQJ]^JhB6OJQJ^Jh 6OJQJ^J(j&hKh06EHOJQJU^J1jO h0OJQJUV_H mH nH sH tH hK6OJQJ^JjhK6OJQJU^J222$2p22222222222"3$3&34363Ĵě{{m\N= hQh(OJQJ^JmH sH h3%OJQJ^JmH sH hQhf 'OJQJ^JmH sH hf 'h`OJQJ]^Jhu'6OJQJ^J(jlhu'h06EHOJQJU^J1jO h0OJQJUV_H mH nH sH tH jh`6OJQJU^Jh`6OJQJ^JhUtOJQJ^Jh`OJQJ^JhMryh`6OJQJ]^JhUt6OJQJ]^J2222"3$3&363MD $Ifgd(kd$$Ifl0b&J  t 0644 lap ytf ' $IfgdC h$If^hgd` & F-$Ifgd`63z3333.4/44466(8)8XYY\]^l $$Ifa$gd(K$ $Ifgd(K$$If[$\$gd( $Ifgd( $Ifgd(63z3333)8XY\]lzĵɵ͵εﲡkVkVkVkVkV(hQh(OJQJ^JmH nH sH tH +hQh(OJQJ\^JmH nH sH tH #hQh(5OJQJ^JmH sH h&ROJQJ^JmH sH  hQh(OJQJ^JmH sH hQh(5OJQJ^JUhQh(<OJQJ^J"hQh(0J5OJQJ\^JhQh(OJQJ^J hQh(CJOJQJ^JaJ!The Greens were the other party whose polling results differed from the ballot box count. The November Roy Morgan poll had the Greens on as much as 10 per cent while others including the Three News TNS and One News-Colmar Brunton had them on nine per cent. However, the disparities largely disappear when the margin of error is taken into account. Actual result Final Herald Digi-poll National Party 45.45 47.9 Labour Party 33.77 36.4 Green Party 6.43 5.8 New Zealand First Party 4.21 3.9 Mori Party 2.24 2.3 ACT New Zealand 3.72 1.8 United Future 0.89 0.7 Jim Anderton's Progressive 0.93 0.2 Number sampled 2,103,842 981 Reported margin of error 3.2% * The poll of 981 respondents was conducted between October 29 and November 5 and has a margin of error of 3.2 per cent. The undecideds were 6.2 per cent.      Mathematics and Statistics in the New Zealand Curriculum Second Tier Strand: Statistical thinking Thread: Statistical literacy Level: Eight k`RR $$Ifa$gd(K$ $Ifgd(K$kd;$IfK$L$lF)  $ ` t06    44 layt(mbTT $$Ifa$gd(K$ $Ifgd(K$kd$IfK$L$lF)  $ ` t06    44 layt(ĵɵ͵mbTT $$Ifa$gd(K$ $Ifgd(K$kdA $IfK$L$lF)  $ ` t06    44 layt(͵εmbTT $$Ifa$gd(K$ $Ifgd(K$kd $IfK$L$lF)  $ ` t06    44 layt(mbTT $$Ifa$gd(K$ $Ifgd(K$kdI!$IfK$L$lF)  $ ` t06    44 layt(mbTT $$Ifa$gd(K$ $Ifgd(K$kd!$IfK$L$lF)  $ ` t06    44 layt( .378SX\]luvz{789:;=±±±פ±פuhW hQhCJOJQJ^JaJhQh9?+OJQJ^JhQh2OJQJ^J hQh(OJQJ^JmH sH h h(CJOJQJ^JaJhQh(OJQJ^J hQh(OJQJ^JmH sH (hQh(OJQJ^JmH nH sH tH +hQh(OJQJ\^JmH nH sH tH #hQh(5OJQJ^JmH sH  .37mbTT $$Ifa$gd(K$ $Ifgd(K$kdQ"$IfK$L$lF)  $ ` t06    44 layt(78SX\mbTT $$Ifa$gd(K$ $Ifgd(K$kd"$IfK$L$lF)  $ ` t06    44 layt(\]lvzmbTT $$Ifa$gd(K$ $Ifgd(K$kdY#$IfK$L$lF)  $ ` t06    44 layt(z{mbTT $$Ifa$gd(K$ $Ifgd(K$kd#$IfK$L$lF)  $ ` t06    44 layt(78mddW $Ifgd  $Ifgd(kda$$IfK$L$lF)  $ ` t06    44 layt(89:;=>@ACDFpni]nnnnnn [$\$gdgd'@kd$$$IflO0b&J  t 0644 lap yt n =>?ABDEGHJϾhebhSWhsiOJQJmH sH  hQhsiOJQJ^JmH sH #hQhsi5OJQJ^JmH sH hsijhsiU#hQhO3<5OJQJ^JmH sH FGIJ$a$gdSW  9r gdlR $ 9r a$gdSW $ 9r a$gdlR$a$gdlR 21h:plR. A!5"W#$% DyK ?http://www.censusatschool.org.nz/2009/informal-inference/WPRH/yK http://www.censusatschool.org.nz/2009/informal-inference/WPRH/yX;H,]ą'cDyK Rhttp://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdfyK http://www.stat.auckland.ac.nz/~teachers/2007/S3MarginOfError/margin-of-error.pdfyX;H,]ą'cMDd b = c $A? ?3"`?2fBGZ갥FWgs4`!kfBGZ갥FWg. d9xڕK@ǿ.h RAVŊSECbK%SW;t(tqQ`{w ̓|w"PlA )*Ais΀vWTyHb_XCmՓbYU8r. 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