Introducing Sample Proportions


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1 Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO, reported that 40% of respodets thought of climate chage i terms of atural temperature variability, rather tha i terms of humaiduced temperature chage. (Referece: How reliable is this proportio, which is based o a radom sample? Does this statistic reflect the proportio of the etire Australia populatio with this belief about climate chage? I this activity, you will ivestigate how much we ca expect proportios from radom samples to vary from sample to sample. Simulatig radom samplig for attitudes to climate chage Ope the TINspire documet Sample_proportios. You will be usig this documet to geerate radom samples from a large populatio. To esure that your results are ot idetical to those of other studets, you will seed the radom umber geerator, as follows. Navigate to Page.. I the Math Box, after the RadSeed commad, iput a space followed by a umber uique to you such as the last 4 digits of your phoe umber. Press eter to execute the commad. Questio Why do you thik that you might get idetical results to those of other studets i the room, if you do ot seed the radom umber geerator of your hadheld? The techology uses a algorithm (rule) to geerate pseudoradom umbers. The algorithm will geerate the same umbers for hadhelds with default factory settigs. Seedig with a umber uique to you will iitialise or reset the radom umber geerator..0 Drawig radom samples of size 50 I this sectio, you will simulate drawig radom samples of size 50 from a large populatio. You will use the sample results to estimate the proportio of the populatio who believe that climate chage is due to atural temperature variability (hereafter referred to as atural climate chage). Let deote the sample size; therefore, i this sectio, 50. Let p deote proportio of the populatio who believe i atural climate chage. Let X deote the umber, i radom samples, who believe i atural climate chage, ad let x deote the values of the variable, X. Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
2 Itroducig sample proportios Aswers & Teacher Notes Let ˆP deote the proportio, i radom samples, who believe i atural climate chage, ad let ˆp deote the values of the variable, ˆP. The values, ˆp, are estimators of the true populatio proportio, p.. Numerical represetatio of sample proportios: = 50 Navigate to Page.. Adjust the slider value to k, to simulate drawig a sigle radom sample from a large populatio, where the value of the populatio proportio, p, is ukow to you. The umber of people, x, i the sample, who believe i atural climate chage is displayed. Note that the values of x are displayed i the spreadsheet (i the colum amed x ) ad as a list i the lefthad pael. Questio a. I the Math Box idicated, use the observed value of x to calculate a poit estimate of the proportio of the populatio who believe i atural climate chage. x, where x ca be the variable symbol, or its value. The computed value is the poit estimate. 50 b. How likely do you thik it is that the poit estimate, calculated above, eds up beig idetical to the true populatio proportio? It is possible, but ulikely, that the poit estimate is exactly equal to the true sample proportio. O Page., press /+e util the spreadsheet pael of the split scree is selected. Press /+R to simulate drawig a differet sample of size 50 from the populatio. Questio 3 a. Use 3 more observed value of x, obtaied after pressig /+R, to calculate 3 more poit estimates of the proportio of the populatio who believe i atural climate chage. It is likely that each poit estimate has a differet value. b. How much variability is there betwee poit estimates obtaied from the differet samples? Aswers will vary. c. Isert a ew Mathbox (/+M) i the lefthad pael. I this Math Box, calculate the mea of the four poit estimates. mea x, or similar, to compute the mea sample proportio. 50 d. Explai why the mea of the four sample proportios is likely to give a better estimate of the populatio proportio tha the idividual poit estimates. It is likely that some poit estimates are greater ad some less tha the true populatio proportio. The mea should balace the overestimates ad uderestimates Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
3 Itroducig sample proportios Aswers & Teacher Notes 3. Simulatig multiple samples: = 50 O Page., adjust the slider value to k 0, which simulate drawig 0 radom samples of size 50. The umber of atural climate chage believers i each simulated sample is displayed i the list ad i the spreadsheet. I the Math Box from Questio a, calculate the sample proportios by iputtig the expressio: 50 x. Store these proportios as a variable, ˆp ; i.e. press Ë(/+h), the Ð(/+k) ad select symbol, ˆp. Press to execute the calculatio. Questio 4 Compare the largest ad smallest umber of atural climate chage believers i the samples, ad their correspodig sample proportios. Calculate the percetage differece betwee these two proportios (i.e. the magitude of the differece, divided by the maximum value ad multiplied by 00). 00 pˆ pˆ Calculatio of the form, pˆ ˆ p. pˆ I the Math Box from Questio 3c, calculate the mea value of all the sample proportios, by iputtig mea p ˆ, ad pressig to execute the calculatio. the expressio: Icremetally icrease the umber of samples draw by chagig the value of k to 40, 60, Questio 5 Do you otice ay patter i the mea of the sample proportios, as the umber of samples, k, icreases? Aswers will vary. Patter ulikely, with the mea likely to fluctuate.3 Graphical represetatio of sample proportios: = 50 Navigate to Page 3.. I Problem 3 you will simulate drawig samples of size 50 from a large populatio where the populatio proportio of atural climate chage believers is kow to be p 0.4. Of course, i practice we would ot carry out samplig if we already kew the value of the populatio proportio. I this activity, the purpose of samplig is to uderstad how samplig behaves. O Page 3. you ca view umerical represetatios of the simulatios, similar to that of Problem. However, i the pages that follow, you will see the graphical represetatio of these results. Navigate to Page 3.3 ad adjust the slider value to k, to simulate drawig a sigle radom sample of size 50, from a populatio where p 0.4. The umber of people, x, i the sample, who believe i atural climate chage is displayed graphically. Adjust the slider value to simulate drawig, 3, 0 samples from this populatio. The vertical lie idicates the mea value. Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
4 Itroducig sample proportios Aswers & Teacher Notes 4 Questio 6 a. For your 0 samples ( k 0 ), what are the observed maximum, miimum ad mea umber of atural climate chage believers? Aswers will vary. b. I a sample of 50 people, how may atural climate chage believers would you expect, if the populatio proportio is 0.4? Why is t this expected umber obtaied each time you draw a sample? Expected umber = There is variability betwee samples is the expected log ru sample proportio. c. Adjust the slider value to k 00. I how may of the 00 samples was the observed value of x equal to the expected value? Aswers will vary..4 The sample cout as a radom variable I the precedig activities, you have see that the value, x, couts the umber of atural climate chage believers i a sample. The cout ca therefore be cosidered as radom variable X, whose values, x, vary from sample to sample. Furthermore, X is biomially distributed, with parameters (sample size) X Bi, p. ad p (populatio proportio); that is Questio 7 Assume that coutig a atural climate chage believer deotes success. Explai why X is a biomial radom variable, with the populatio proportio beig the probability of success i a sigle trial. A sigle trial cosists of determiig whether a radomly chose idividual believes i atural climate chage. X is biomial because the idividual either believes i atural climate chage ( success ), or they do ot ( failure ).5 The sample proportio as a radom variable Navigate to Page 3.4, ad icremetally icrease the umber of samples observed, by adjustig the value of k, up to k 00. The observed sample proportio for each of the k samples is displayed graphically. The vertical lie shows the mea of the sample proportios. If you avigate back to Page 3.3, you will observe a aalogous graphical patter for the success cout, for the samples of size 50. Questio 8 a. Explai why the graphs i Pages 3.3 ad 3.4 have idetical shapes. The two graphs have idetical shapes because the two quatities, x ad ˆp are related by a costat ratio b. For a particular sample, what is the relatioship betwee the sample cout of successes, x, ad the sample proportio of successes, ˆp. ˆp x Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
5 Itroducig sample proportios Aswers & Teacher Notes 5 You will have observed that each time you take a radom sample from a large populatio, the sample cout of successes, x, ad the sample proportio of successes, ˆp, vary from sample to sample; x furthermore, for a particular sample of size, ˆp. Just as the cout ca be cosidered a radom variable X, whose values, x, vary from sample to sample, the sample proportio ca, likewise, be cosidered a radom variable ˆP, whose values, ˆp, vary i the same fashio as x. Therefore, cosiderig the radom variables, ˆP X.. Expectatio ad stadard deviatio of the sample proportio Navigate to Page 3.5, ad icremetally icrease the umber of samples observed, by adjustig the value of k, up to k 00. The observed frequecies of sample proportios for the k samples is displayed as a histogram. Navigate to Page 3.6 to view the same data, displayed as a boxplot. From these, we observe that the sample proportio has a distributio, for which a mea ad stadard deviatio ca be computed. Below we will cosider the theoretical expectatio (mea) ad stadard deviatio of the sample proportio. Questio 9 Recall that ˆP Further, if Y X ax, where X Bi, p. Also, for a biomial radom variable, E X, where a is a costat, the EY ae X a. Usig the above iformatio, show that EPˆ E ˆ X P E E X p E Pˆ p p. p. b. Explai the sigificace of the result obtaied i part a. above. The mea of the distributio of sample proportios gives the populatio proportio. As the distributio is cetred at p, i the log ru, o average, the sample proportio will either overestimate or uderestimate the value of p. Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
6 Itroducig sample proportios Aswers & Teacher Notes 6 Questio 0 For a biomial radom variable, var X p p. Further, if Y ax, where a is a costat, the var Y a var X. a. Usig the above iformatio, ad Questio 9, show that the stadard deviatio of the sample proportio, SDPˆ var Pˆ p p. X var var X p p p p b. The variace ad stadard deviatio of ˆP have i the deomiator. Explai the implicatios of this, i terms of the spread of the distributio of ˆP. A key implicatio is that as (large sample size), ˆ SD P 0 (the spread of the distributio of ˆP is small). Sice the distributio is cetred at p (see Q. 0b), a small spread makes it more likely that the sample proportio is close to the true value of the populatio proportio. The cocepts itroduced i this activity are explored further i the activity titled Distributio of sample proportios. Texas Istrumets 05. You may copy, commuicate ad modify this material for ocommercial educatioal purposes provided all ackowledgemets associated with this material are maitaied. Author: Frak Moya
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