Are Smiles Contagious?

Josh Combs

Trevor Combs

Viktoria Safarian
5/27/08 B3

Disclaimer

This study was done in an AP Statistics course with relatively small sample sizes. The validity of such studies must always be questioned. Please keep this in mind if you use or report the results of this study.

Abstract

The goal of this observational study was to determine if smiling at a stranger in a shopping mall would induce the stranger to smile back. Specifically, the question posed was “is there significant evidence to prove that smiling is contagious?” The hypothesis was that smiling is, in fact, contagious and a significant proportion of people (more than half) would smile back. The reasoning behind this expected outcome was that it is polite and culturally conventional in our society to smile back at people who smile at you. The population was Fayette Mall shoppers on Tuesday, April 13, 2008 and Wednesday, the 14th. We used systematic sampling. The sample was every fifth subject who made eye contact with the observing researcher (the observing researcher smiled at 300 subjects, but only 137 looked back at him so the 163 no-responses were not included in the smile/no smile proportions). The test performed on the raw data was the one proportion z test of significance.

Methodology

The saying goes “smiles are contagious.” But, are they really? Does smiling at a person prompt the person to smile back? The question is general and sounds simple enough, but the stipulations and conditions of the sampling process make it more complicated. Walking through the entire mall eliminated locational bias (different types of shoppers located around different stores). The observing researcher (a.k.a. Josh Combs) walked through Fayette Mall and, using systematic sampling, smiled at every fifth person that he saw. He randomly chose the first person to sample by walking through the mall for two minutes and smiling at the first person that he saw after two minutes. He smiled at every fifth person because he wanted to avoid bias from big groups walking together and so that he could have time to record the results and look back up to the next subject. This was the treatment imposed. His smile was the same for every subject to make for the most consistent treatment. If the subject made eye contact and smiled back, Combs, with his cell phone handy, typed in the number 1. If the subject made eye contact and did not return the smile, Combs typed in the number 2. If Combs recognized the fifth person or if the fifth person did not make eye contact, Combs typed in the number 3. The subjects that correlated to the number 3 were not accounted for in the total of the sample because we defined our population to be the people who actually saw the smile and could therefore “catch” the “contagious” smile. However, these people were not completely disregarded. Throwing out their data would have been bad statistical sampling because they had the potential of introducing non-response bias. So, instead of including them in our sample, we simply note them in our report. The total population of shoppers at Fayette Mall on a regular Tuesday night was estimated, by the amount of cars in the parking lot and approximately two or three people per car, to be around 2000 – 3000. The sample size was determined from the size of the population. Because the population was approximated at 2000 – 3000, which had to be at least 10 times the sample size, the sample size was 240. Another trial, on late Wednesday night, produced a smaller sample because the mall was less crowded. The total number in the sample was 300. Because many of the subjects (163) in our sample did not look up and make eye contact, there were many non-responses. These people are not in our population because if the person does not perceive a smile, the smile cannot be contagious. The subjects that did not make eye contact were excluded from the sample. Our actual sample size was 137.

Explanatory Data Analysis ^*

	SMILE	NO SMILE	NO RESPONSE	TOTAL
count	41	96	163	300
p (out of 137 responses)	0.299927	0.7007	N/A	1

Analysis and Inference
Assumptions:
Sample is an SRS √ *random sample (systematic)
Population is at least ten times the size of the sample √

np_o ≥ 10 137(.5) = 68.5 > 10 √

n(1-p_o) ≥ 10 137(1 - .5) = 68.5 > 10 √

Hypotheses:

H_o P_smile = 0.5

H_a P_smile > 0.5

Test of Significance:

z =

z =

z = -4.69897

P (z ≥ -4.69897) = .9999

Conclusion

Because the p-value is .9999 which is well above the α = .05 significance level, the data is not statistically significant and we must retain H_o. This indicates that the proportion of people who do not smile back when they see a stranger smiling at them in a public mall is not greater than .5. Because of the extremely high P-value, a viable observation would be that the proportion of strangers who would not smile back is actually greater than the proportion of strangers who would (P_{not smile} > 0.5). This might warrant another significance test, with the alternative hypothesis that more than 0.5 strangers would not smile back.

This study had several weaknesses that might have affected the proportions. One of these was the extremely high number of subjects who did not respond to the treatment. The 163 non-responses were people who simply did not make eye-contact with the observing researcher. This is about 54.333% of the subjects. This high proportion could have swayed the results in one direction or another. Although, we cannot draw conclusions from this stipulation, we can define our population as those people who actually look back at the observing researcher. Another weakness was our hypothesis. The alternative hypothesis was “very statistically insignificant”. A possibility was that more than half of those sampled who responded to treatment would not have smiled back. However, this unpredictability is an integral part of experimental statistics and we must learn from our mistakes.

Appendix

*The raw data was recorded in a cell phone text message and we could not include it in our report. However, the totals are all summed up the table.

*We could not randomize the sample because of the nature of our population. Systematic sampling was used. This might introduce some bias.