Question

AP Statistics-The Great Marshmallow Experiment An Investigative Task in Behavioral Economics and Psychology The experiment was conducted in 3 phases. In the first two phases, experimenters promised children better art supplies (better crayons, then better stickers) for a fun project if the child waited a specified period of time; since the researchers wanted the children to wait, the original supplies were packaged securely and all children ended up waiting the intended time period However, half the children were actually rewarded with the better supplies, while the other half were provided merely with an apology and no improved supplies. Then, the marshmallow test was presented as a snack time activity. Would the children who had reliably been rewarded with the better art supplies as promised be more likely to wait for the second promised marshmallow than those children who had their promises of better art supplies unreliably reneged upon? Your current assignment: investigate the hypothesis posed above. Write up the statistical procedures you used and your conclusions (in context, of course) Use the sample data below to create a 95% confidence interval for the mean wait-time for the children in the 'reliable' group as well as f Based on your confidence intervals, compare the performance of each of these groups of children to the mean wait-time in the original experiment (descrnbed in the background section above or the unreliable group - - Run a one-sample i-test for means to compare the reliable group to the claimed mean in the original study Compare the proportion of children who waited 15 minutes in each group (using either a 2 proportion hypothe sis test or a 2 proportion confidence interval). Compare the results and methods of this statistical procedure to the previous three Subject Gender WaitWaited 15 SubjectGede WaitWaited 17 785 431 15 10 10 12 14 16 145 498 457 73 20 AI 594 150 195 150 24 26 28 AI AI 27 AP Statistics-The Great Marshmallow Experiment An Investigative Task in Behavioral Economics and Psychology Background: Much literature is available, regarding delayed gratification, documenting the original famous 1974 "Marshmallow Experiment" in which young children were offered a laboratory choice- task: eat one marshmallow immediately or resist the temptation for an extended delay and be rewarded with two marshmallows. The majority of children, 75%, could not wait the required 15 minutes to earn their second marshmallow; the mean wait time was reported as 5 minutes 6 seconds. The researchers followed these children into adolescence and reported various revealing findings. For example, ฯ0nger wait-times among children were correlated with greater self- confidence and better interpersonal skills, according to parental report. Longer wait-times also correlated with higher SAT scores (Shoda et al, 1990), less likelihood of substance abuse (Ayduk et a., 2000), and many other positive life outcomes (cg. Mischel, Shoda, & Rodniguez 1989) Based on these findings, the marshmallow task was argued to be a powerful diagnostic tool for predicting personal well-being and later-life achievement-an carly indicator of an apparently long-tem personal quality" One possible explanation for a child's inability to wait is a lack of self-control; interpreted this way, the research would suggest that self-control predicts future success Question: Yet, what if something else was driving the child's failure to delay-of-gratification? More specifically, what if the child's beliefs and expectations of the world around them shaped their willingness and ability to delay gratification in the promise of larger future reward? Under this theory, children make rational decisions based upon beliefs that the child acquired before entering the testing room. "Waiting is only the rational choice if you believe that a second marshmallow is likely to actually appear after a reasonably short delay and that the marshmallow currently in your possession is not at risk of being taken away. This presumption may not apply equally to all children." Children living with absent parents or living in shelters may behave quite differently based on life circumstances and leamed expectations. "For a child accustomed to 돼olen possessions and broken promises, the only guaranteed treats are the ones you have already swallowed." It is quite possible that "children's willingness to wait is negatively impacted by uncertainty about the likelihood, value, or temporal availability of the future reward." To test the hypothesis that the reliability of the experimenter in the testing environment influences children's wait-times during the marshmallow task, researchers conducted a modified marshmallow experiment with 28 children aged 3-5 years old in which half of the children observed evidence that the researcher was reliable in advance of the marshmallow task, while half observed evidence that she was unreliable; the two experiment groups were assigned randomly. The researchers surmised that if children employ a rational process in deciding whether or not to eat the first marshmallow, then the children in the group that have evidence that the adult is a reliable person to be significantly more likely to wait than those in the unreliable condition. Kidd, Palmeri, Asin, Rational snacking: Young children's decision-making on the marshmallow task is moderatang by beliefs about ㎝vironmental reliability. Cognition (Volume 21, Issue l. January 2013, pp 109-114) Source: htp://www.sciencedirect.com/science/article/pis0010027712001849

Answer 1

For the fourth part , test of significance for difference of proportion is asked. But for the test of significance for difference of proportion the sample sizes should be large (greater than 30 ) as for large samples the proportion difference is asympotically normal distributed. Here the sample size is 14 so this test cannot be applied.

The 95% Confidenu mtewa, bor mean whm Ales than 30 (small) where x ia the. sample mean, tx unbiased estimat, of poph vawiance tn-(%) ''s the crtical value at dlud ot tieedem and n u the si?e of For Reliable ar ΣΧίζ= 8469323 14 1162685こ39437-340 13 87437' 3 yo 2.1603 a99, 0607 s Tabulld t 14-1 (0.05) 土 2.1603 X 299, 0607 722428 722.428 i 172.6725 549.156 , 395.10") 's tuv 95% onfidenu intewat or the mean wait time fot chddvn um te Heliaa goup

→ Vhnliable group 2 e01357346(181 571* 13 -2-1 h- 896041-Y29-68926.,2637 Ч 13 s= 68926 . 26374 = ằ62.53 8,18ί tabuldtnl)- 2.1 603 101.57/y ± 2.1603 X 2625381186 29.98643 , 333.ISCY | ui thu 95% confidine intewat r the mean

3 Mean wad of original expoumendis : 5 mun b see 300 t 6 306 econds Ho : μ= 306 seconds ie., the mean wart tivme was 306 secemds Now if the confidence intewal does ot contaun the mul ditFesene between the samble mean lu o thesis value than tha sinifcant emean χ and -Ą For reliable grơuf 4.7561 85.loli) do not contain the fheiz cant diterent betuueen the. le grevp and thesus value. T mul 13 SI → For urtdiasle A4.q0643 , 333-I564 contauns the null igmi frand dlenc Letwesn tha 8I

In) Ohe »ampla t-tut to uompaxe the veliable tha daimed mean ui the omal stu oup to Ho : There iA no initicant diffeanu betwun vwait time of reli able the mean and the cdaimed mean un o St ie., 从 3,06 secondう Hi Thae is sinit'cant dittaene betweer the mean wait tum of lable group and tha damed mean in on Undai Ho In paxt U) 1 we have thei value Usinq them y xこ722429 S= 299,0607 ulated 30 6 tiy(%)-2.16036. tay: (722.428-306)- 29 0607 - I558.13090 294. 06 01 tea/ 5.210 As the calculated valiue ot t geater than the tabulated value oft we have enough evideno to Sl SI ect the null hToth s at s7e Level ot cane and we may condude that thase ui cant du u between the mean wat tmą and the daumad mean un of st он