Introduction

The research question of this study is: Is there an association between wearing glasses and people’s intelligence level?

The population parameter of interest in this study is the average intellectual level difference of people who wear glasses compared to people who do not wear glasses. The researcher is interested with this research topic is because he read the article “The effect of wearing eyeglasses on the perception of attractiveness, confidence, and intelligence” which talked about public’s feeling of wearing glasses people’s photos in the standard of attractiveness and intelligence. The content discussed the stereotype that people who are wearing glasses are less attractive and more intelligent public holds. The result is people in different culture are having different stereotype toward glass wearing people with intelligent level, and this means for some culture, public do not believe that people wearing glasses because they read a lot which makes glasses wearing a smart looking characteristic. Therefore, the researcher wants to compare the intelligence level between glass wearing group and non-glass wearing group to figure out if the stereotype glass wearing people are more intelligent correct or not. Before read any data about people’s average intellectual level, the researcher suppose the average intellectual level of glasses wearing people are higher than others who do not wear glasses.

Data Collection Methods

The observational units are island dwellers. The variables are whether the dweller wearing glasses or not(binary categorical variable), and his/her IQ score getting from IQ test(quantitative variable).

The researcher observed that there are circle towns and square towns on the island’s map, and people dwell in a circle town are having higher possibility of having an higher IQ, higher income, and higher possibility of wearing glasses. Therefore, when measured the data, researcher first uses a random generator to first randomly pick the one circle town and one square town from each island, and use the random number generator to pick about 10 people from each of these 6 towns. If someone refused to participate in the study, the researcher will randomly find another tester until there are at least ten people from this town. As a result, the data can generalize the intelligence level people from all three islands exclude the factor of geological location. The sampling error might exist because I am choosing similar amount of people from circle towns and square towns, however, the people living in a circle town and others who live in square towns are having different average value and the population size of these two kind of towns are largely different either. As a result, choosing similar amount of people from circle towns and square towns might not represent the actual average intellectual level of people on the island. However, this will not effect the comparison between the intellectual level of glasses wearing people and non-wearing people, because each town contains either glass wearing people and non-wearing people and the study is using random sampling, so everyone in each individual town no matter their intellectual level and glasses wearing condition are having equal opportunity as everyone else in the same town.

Descriptive Statistics

boxplot( `IQ test score` ~`Wearing glasses or not`, 
       horizontal = TRUE, 
       main="Side-by-side boxplots",
       data = Data)

favstats(`IQ test score` ~ `Wearing glasses or not`, data=Data)

Wearing glasses or not is a binary categorical explanatory variable, and IQ test score is a quantitative variable. The glasses wearing group are having a median intellectual level of 110 which is significantly higher than the non-wearing glasses group’s median IQ test score 102. The mean value, maximum value and minimum value of average IQ score of the glasses wearing group are all greater than the non-wearing glasses group. Overall, the IQ score mean difference between glasses wearing gorup and glasses non-wearing groups 12.44. which seems to be a gigantic difference. Moreover, on the graph, we are able to see that the average range of IQ score of the glasses wearing group is higher than the none-wearing group, as a result of the graph and the summary statistics，from the side-by-side box plot, we are able to say there is an association between the two variables.

Analysis of Results

The population in the study are all the islanders in the island, with all age group included. The null Hypotheses of this study is there is no difference in average IQ between people wearing glasses and people that do not wear glasses. \(H_{0}: \mu_{glasses} - \mu_{non-glasses} = 0\) Contrary, the alternative hypothesis of this study is there is a difference in average IQ between people wearing glasses and people that do not wear glasses. \(H_{a}: \mu_{glasses} - \mu_{non-glasses} > 0\)

The type I error in this study is there is no difference in average IQ between people wearing glasses and people not wearing glasses(null hypothesis is true), but this study rejects the null hypothesis. The type II error will be there is a difference in average IQ between people wearing glasses and people not wearing glasses(null hypothesis is false), but this study fails to reject the null hypothesis.

The measurements in the data are all using random sampling, that all the tester are picking by a random generator. Moreover, the study is choosing same amount of people from two random towns on all three islands, so the data get from the measurement should represent all the on population on the island with no age and gender limitation.

Theory-Based Analysis There is a theory-based analysis approach below, however, because there are only 16 testers in the glasses wearing group of this study, and glasses wearing group’s IQ distribution in the side by side boxplot above is right skewed. Therefore, the validity condition of the study is failed, and the researcher will also perform a simulation-based analysis on this study.

#t-statistic
stat(t.test(`IQ test score` ~ `Wearing glasses or not`, data = Data))
##        t 
## 2.606944

The t-statistic is 2.60, which means observed difference in IQ average between glasses wearing group averages and glasses not wearing groups is 2.606944 standard deviations above zero in the null hypothesis.

#p-value
pval(t.test(`IQ test score` ~ `Wearing glasses or not`, data = Data))/2
##     p.value 
## 0.006974476

The right-side p-value is 0.007 which means that there is only 0.7% probability assuming that the null hypothesis is true that there there is no difference in average IQ between people wearing glasses and people that do not wear glasses. The right-side p-value is 0.007 smaller which is smaller than 0.05 also means that the study shows very strong evidence against the null hypothesis. Since is the study shows shows very strong evidence against the null hypothesis, the study shows very strong evidence accepting the alternative hypothesis that there is a difference in average IQ between people wearing glasses and people that do not wear glasses.

Based on the 2.60 t-statistic and 0.007 right-sided p-value, both shows very strong evidence against the null hypothesis, the researcher is ability to conclude that the null hypothesis in this study based on the collected data is being rejected, and the alternative hypothesis is begin supported by the data. Back to the research question, this means that there is a positive correlation between wearing glasses and intellectual level.

Simulation-Based Approach

set.seed(1)
Data.null <- do(1000) * diffmean(shuffle(`IQ test score`)  ~ `Wearing glasses or not`, data = Data)
dotPlot(~ `diffmean`, data = Data.null,
main="Simulated Null Distribution of the difference in sample means",
xlab="difference in sample means",
width = 0.5, cex = 1
,groups = (diffmean >= 12.44388))

p_value<-prop(~(diffmean >= 12.44388), data = Data.null)
cat("The simulation-based two-sided p-value is",p_value)
## The simulation-based two-sided p-value is 0.007

Even though the validity condition for my theory based approach is not valid, the one-sided p-value 0.008 of simulation approach is similar with the theory-based approach’s right-side p-value 0.007. These two p-value are similar which strengthen the previous conclusion about this study strongly against the hypothesis that there are no difference between average IQ level between glasses wearing gorup and not wearing glasses group, which also means approaches provide strong evidence supporting the alternative hypothesis.

confint(t.test(`IQ test score` ~ `Wearing glasses or not`, data = Data))

The 95% confidence interval is between the lower bond 2.70 and upper bond 22.18, and it does not include the possibility that the difference mean between glasses wearing people’s IQ score and non-glasses wearing people’s IQ score is 0. Which means that when we are 95% confident that the long run difference mean is between 2.70 and 22.18, but the null hypothesis is not in one of these possibilities. This result is also corresponding with the theory-based approach and simulation approach conclusion.

Conclusion

In this study, the researcher learns that there are correlations between wearing glasses and intellectual level, and it is a positive correlation, which means the conclusion that “glasses wearing people are averagely have higher average intellectual level level comparing with not wearing glasses people” can be made based on this specific study’s statistic analysis.This difference mean between two groups average IQ score is larger than the researcher originally conjectured. The researcher originally think this might be a small difference between the average intellectual level of the two groups, but the result actually shows there is a large gap. If there are some revision’s can be made during the data collection part, the researcher wants to randomly collect more samples, specifically randomly choose 1% people of people in each of the six town I choose. Therefore, there be a larger sample size, and the average IQ score can be more representative to the whole population on the island.

Bibliography: references to literature mentioned in the introduction

Williams KM, Hysi PG, Yonova-Doing E, Mahroo OA, Snieder H, Hammond CJ. Phenotypic and genotypic correlation between myopia and intelligence. Sci Rep. 2017 Apr 6;7:45977. <doi: https://doi.org/10.1038/srep45977>

Literature reference:AlRyalat, S. A., Mohammed, J., Al Hajaj Sari, W., Al-Noaaimi, F., Yazan, A., & Al-Rawashdeh, A. (2022). The effect of wearing eyeglasses on the perception of attractiveness, confidence, and intelligence. Cureus, 14(3) doi:https://doi.org/10.7759/cureus.23542