# Chi-Square. Hypothesis: There is an equal chance of flipping heads or tails on a coin. Coin A. Expected (e) (o e) (o e) 2 (o e) 2 e

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1 Why? Chi-Squar How do you know if your data is th rsult of random chanc or nvironmntal factors? Biologists and othr scintists us rlationships thy hav discovrd in th lab to prdict vnts that might happn undr mor ral-lif circumstancs. At som point thos prdictions ar ithr supportd or not by ral data. It would b nic if th two sts of data matchd xactly, but mor oftn than not thy don t. This could b attributd to random fluctuations in th variabls, or it could b du to a factor that th scintists ovrlookd. How dos th scintist know? Thr ar many statistical calculations that hlp answr this qustion. On that is commonly usd by biologists is Chi-Squar. Modl 1 Calculating Chi-Squar (χ 2 ) Hypothsis: Thr is an qual chanc of flipping hads or tails on a coin. Coin A Obsrvd data (o) Expctd () (o ) (o ) 2 (o ) 2 χ 2 = Σ(o ) 2 Hads Tails Total Coin B Obsrvd data (o) Expctd () (o ) (o ) 2 (o ) 2 χ 2 = Σ(o ) 2 Hads Tails Total What is th hypothsis that has bn tstd in Modl 1? 2. Dscrib in on or two complt sntncs th xprimnt bing prformd in Modl How many flips of th coin will b conductd in ach trial for th xprimnt in Modl 1? Chi-Squar 1

2 4. How many of th 200 flips would you xpct to b hads and how many would you xpct to b tails? Fill in th xpctd () column of Modl 1 for both coin A and coin B. 5. Assum th xprimnt for coin A rsultd in 108 hads and 92 tails. Fill in th data tabl for coin A undr obsrvd (o). 6. Th xprimnt was rpatd with a diffrnt coin. coin B rsultd in 120 hads and 80 tails. Fill in th data tabl for coin B undr obsrvd (o). 7. If you wr told that on of th coins usd in th xprimnts in Modl 1 was a trick coin, which coin would you prdict was riggd? Explain your rasoning. Rad This! Th xprimnts in Modl 1 did not giv th xpctd outcom, so th qustion bcoms: Is this du to random chanc, or dos th coin bing flippd favor hads for som rason? Th statistical calculation of chi-squar (χ 2 ) will hlp dtrmin if thr is a significant diffrnc btwn th xpctd outcom and th obsrvd outcom. In statistics a significant diffrnc mans thr is lss than 5% chanc that th variation in th data is du to random vnts. Thrfor, th variation is most likly du to an nvironmntal factor. 8. What symbol rprsnts chi-squar? 9. Us th quations in Modl 1 to complt th calculations of χ 2 for coin A and coin B. 10. Circl th corrct phras to complt th sntnc. A largr chi-squar valu mans th obsrvd data is (vry diffrnt from/vry similar to) th xpctd data. 11. On hundrd htrozygous (Bb) mals mat with on hundrd htrozygous (Bb) fmals. a. Draw a Punntt squar to show th possibl gnotyps of th offspring from ach pairing. b. Prdict th numbr of offspring from th 100 mating pairs that will b ach gnotyp. BB = Bb = bb = 2 POGIL Activitis for AP* Biology

3 c. Th obsrvd outcom from this xprimnt is 28 BB offspring, 56 Bb offspring and 16 bb offspring. Us a tabl similar to that in Modl 1 to calculat chi-squar for this xprimnt. d. In your opinion, is th obsrvd data significantly diffrnt from th xpctd data in this mating xprimnt? Justify your answr. Rad This! To dtrmin if th chi-squar valu is larg nough to b significant, rsarchrs nd on mor thing dgrs of frdom. If an xprimnt has fiv possibl quivalnt outcoms, thn in rality thr is on rsult and four additional possibl rsults, making a total of fiv. Th dgrs of frdom ar always calculatd as th numbr of possibl outcoms minus on. 12. Considr th coin flip xprimnts in Modl 1 whr you could gt hads or tails. a. How many outcoms wr possibl in th coin flip xprimnts? b. How many dgrs of frdom wr thr in th coin flip xprimnts? 13. Considr th mating organisms in Qustion 11. a. How many gnotyp outcoms wr possibl in that xprimnt? a. How many dgrs of frdom wr thr in that xprimnt? Chi-Squar 3

4 Modl 2 χ 2 Analysis Dgrs of Frdom χ 2 Valus P Valu = % probability Not Significant Significant 14. Th tabl in Modl 2 has thr typs of valus χ 2, dgrs of frdom, and P valus or probabilitis. a. Which of ths ar along th bottom of th tabl? b. Which of ths ar along th sid of th tabl? c. What chi-squar valu is ndd to hav a P valu of 0.5 in an xprimnt with two dgrs of frdom? 15. Th tabl in Modl 2 is a rfrnc tabl usd by scintists to intrprt th calculatd chi-squar valu for thir xprimnt. It convrts th chi-squar valu into a probability that th diffrncs in th data ar only du to chanc. a. Which P Valus in th tabl indicat that th diffrnc btwn th xpctd and obsrvd data was likly du to chanc? b. Which P Valus in th tabl indicat that th diffrnc btwn th xpctd and obsrvd data was likly du to an nvironmntal factor? 16. What dos a P valu of 0.7 man in trms of prcnt chanc that th data sts ar diffrnt only by chanc? 4 POGIL Activitis for AP* Biology

5 17. As th chi-squar valus gt largr, is th data mor likly to b significant or not significant? 18. Whn an xprimnt has mor dgrs of frdom, is a largr chi-squar ndd for a significant outcom, or is a smallr chi-squar ndd for a significant outcom? 19. Which row of χ 2 valus in Modl 2 should b usd for th coin flip xprimnts? 20. Find th P valus for ach coin xprimnt in Modl 1 using th χ 2 valus in Modl 2. Not: If th χ 2 valu falls btwn two points on th chart, giv a rang of P valus. a. Coin A = b. Coin B = 21. Considr your answrs in Qustion 20, a. Which coin producd a nonsignificant χ 2 valu, indicating that th outcom was not statistically diffrnt from what was xpctd? b. Which coin producd a significant χ 2 valu, indicating that th outcom was statistically diffrnt from what was xpctd and thrfor du to an nvironmntal factor? c. With your group propos an xplanation for why th rsults of ithr coin wr statistically significant. 22. Dos th mating xprimnt from Qustion 11 indicat that an nvironmntal factor was affcting th outcom? Justify your answr using information from Modl 2. Chi-Squar 5

6 23. A rsarchr is invstigating a flowring plant. Th purpl flowr alll, P, is dominant to th whit flowr alll, p. A cross was prformd btwn a purpl-flowrd plant and a whitflowrd plant. Th 206 sds that wr producd from th cross maturd into 124 purplflowrd plants and 82 whit-flowrd plants. Construct a tabl similar to thos in Modl 1 to calculat chi-squar for th null hypothsis that th purpl-flowrd parnt plant was htrozygous. 6 POGIL Activitis for AP* Biology

7 Extnsion Qustions 24. You hav prformd a dihybrid cross of plants and got th following data: 206 purpl tall, 65 whit tall, 83 purpl short, 30 whit short. a. Using your knowldg of gntics, dtrmin th xpctd outcom for this xprimnt. b. What ar th dgrs of frdom in this xprimnt? c. Calculat chi-squar for your xprimntal data blow. Obsrvd data (o) Expctd () (o ) (o ) 2 (o ) 2 Purpl and tall Purpl and short Whit and tall Whit and short d. Dos your collctd data diffr significantly from th xpctd valus? Explain. Chi-Squar 7

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