Due to the differences in societal conditioning of men and women, there can be differences in the ways that each group presents or perceives themselves. One of the ways this manifests is in confidence levels, which is what led this study to ask the question of whether or not there is a difference between men and women and how much they overestimate their IQ. The subjects of the study were gathered from the fictional population of The Islanders, each person being an Islander. The population parameter of interest in this study was the true mean difference in IQ overestimation between males and females on the Islands. While men and women are considered to be equal matches in intelligence, there may be a trend when comparing how the two groups over or underestimate themselves. In one study conducted on the topic, researchers observed a tendency for underestimation in females and overestimation in females (Reilly et al., 2022). Another study focused on the idea of “self perceived humility” and how that might differ based on the social conditioning of men and women (Szymanowicz & Furnham, 2013). The two studies in conversation with each other led to the formation of the chosen research question for this study. Based on the conclusions of the established literature, that led to the suspicion that the actual mean difference in IQ overestimations would be different from zero, and more likely that the overestimated values for men would be higher than the overestimated values for women.

The observational units in this study were the 120 adults on the Islands that were sampled. Samples of 20 were taken from two towns on each island for a total of 120 people. The towns sampled were Hofn, Helvig, Vaiku, Eden, Arcadia, and Reading. Each of the individuals was chosen randomly via a random number generator based on the population of the town (i.e., if there are 100 people in the town a number between 1 and 100 will be generated). From there, another random number generator was utilized to randomly select a member of the household. The only caveat was that the selected member had to be an adult, as data was taken on the adult IQ scale. The variable of the overestimation of IQ was measured by first asking each subject what they thought their IQ was, and then asking them to take an actual IQ test, and then subtracting their actual score from their estimation. For example, if someone estimated their IQ to be 111, but it was actually 101, the overestimation would be 10 points. During the course of the study, the occasional subject would withdraw from the study when data was being recorded. This would happen about 15% of the time. When this happened, another participant would be randomly selected from the same island and the process repeated. If the selected subject refused to participate, a different randomly selected person would be asked until the desired sample size of n=120 was reached. The withdrawal of certain participants could have led to some sampling error when they were replaced in the study, and there were a few repeat visits. Part of this could be that time passed between when the subjects were asked to take the IQ tests and when they were checked on to collect the final . It’s possible that me waiting multiple hours or days to record the data influenced whether or not the subject withdrew. In the future, it would be recommended that the data is collected as soon as possible.

The variables that were measured in this study were sex of the subject, the self reported IQ of the subject, and the actual measured IQ of the subject after taking an IQ test. The sex of the subject is a binary categorical variable, as the Islanders only respond to the question with male and female. This variable is easy to measure as it can easily be included as the question of sex on a survey. The subjects self reported IQ and actual measured IQ are quantitative variables, as IQ is measured as a number of points. The variable that will be compared to sex, however, is the difference between estimated IQ and measured IQ (estimated-measured), which is a quantitative variable as well. All of these variables are relatively easy to measure—Islanders can be asked what they think their IQ is on a survey and 20 minute IQ tests are readily available through the tasks tab. The more difficult part is manually calculating the difference between estimated and measured IQ. For the statistics themselves, the mean average difference for women was subtracted from the mean average difference for men when seeing the difference between sample means. For example, if the average difference for men is 3 points and the average difference for women is -2 points, then the difference in men minus women is 5 points.

MiniProject3Data <- 
  read_csv("~/MATH247/Data/Jensen Project Data - Sheet2.csv")

bwplot(Sex ~ Overestimation, 
       horizontal = TRUE, 
       main="Sex vs. Overestimation of IQ",
       data = MiniProject3Data)

data(MiniProject3Data)
favstats(Overestimation ~ Sex, data = MiniProject3Data)

Based on the boxplot, there appears to be a very strong association between sex and overestimation of IQ. The max overestimation for women barely overlaps with the minimum overestimation for men at -1 point. Additionally, the mean overestimation for women was -9.5, and the mean overestimation for men was 7.7, which is a 17.2 point difference. The medians are 18 points apart, at -10 for women and 8 for men. The standard deviation of both groups is roughly the same, and 3.82 for women and 3.9 for men. The distribution for both groups appears to be roughly symmetric, with no outliers.

The population being surveyed in this study is the general population of the Islands. The towns that were specifically sampled from are Hofn, Helvig, Vaiku, Eden, Arcadia, and Reading. The parameter of interest is the true mean difference in IQ overestimation between males and females on the Islands. Due to the fact that all the data was randomly sampled and the sample size is relatively large at n=120 with samples from each Island, the measurements can reasonably be considered a representative sample for this population of interest. For this study, we assumed, as our null hypothesis, that there was no association between the sex of an individual and the degree to which they overestimate their IQ. The alternative hypothesis was that there was an association between the sex of an individual and how much one overestimates their IQ.

\[H_0:\mu_{male}-\mu_{female}=0\]

\[H_A: \mu_{male}-\mu_{female}\ne0\] In studies like these, sometimes incorrect conclusions can be drawn about the null hypothesis. The first kind is a Type I error, which is when the null hypothesis is rejected, but the null hypothesis is actually true. In this study, this would be if evidence was found against the null hypothesis, leading us to conclude that there is a difference between men and women and how much they overestimate their IQ (i.e. men overestimate more than women or vice versa), when in actuality, there is no significant difference between the two groups. On the other side of the coin, a Type II error is when the null hypothesis fails to be rejected, but the null hypothesis is actually false. In the context of this study, that would mean that we did not find significant evidence against the null hypothesis that men and women overestimated their IQ differently, but in reality there was a difference.

The validity conditions for a two sample t-test for means are that there are at least 20 observations for each group (in this case males and females) and that there is no strong skew in the distribution of either group. For this study, there were 55 females and 65 males surveyed, and because both of those values are over 20, that validity condition is met. To determine if the other condition is met, we will look at a histogram output.

histogram(~Overestimation | Sex, data = MiniProject3Data, width = 1, layout = c(1, 2))

Since both distributions appear to be roughly symmetric and normally distributed with no strong skew, the second validity condition is met as well.

\[t=\frac{7.7-(-9.5)-0}{\sqrt(3.91^2/65+3.82^2/55)}=24.31\]

The t-statistic that was found for this study was 24.31, which gives very strong evidence against the null hypothesis that there is no association between sex and IQ overestimation.

two.sided.p.value<-pval(t.test(Overestimation ~ Sex, data = MiniProject3Data))
cat("the two-sided p-value is",two.sided.p.value)
## the two-sided p-value is 2.68631e-47

With a p-value of approximately zero, there is significant evidence against the null hypothesis that there is no association between males and females and how much they overestimate their IQ at the 0.05 level of significance. In other words, the probability of observing a 17.2 point difference in IQ overestimation between men and women assuming that the null hypothesis is true is essentially 0%. From this, we make the statistical decision to reject the null hypothesis. This suggests an association between someone’s sex and how much they overestimate their IQ.

x.bar.male <- 7.7
x.bar.female <- -9.5
x.bar.diff <- 17.2

SD.male <-3.82
SD.female <-3.91

n.male <- 65
n.female <- 55

SE.x.bar.diff <- sqrt(SD.male^2/n.male + SD.female^2/n.female)

MoE <- 1.96*SE.x.bar.diff

LB<-x.bar.diff - MoE
UB<-x.bar.diff + MoE
cat("The 95% confidence interval for population mean is",round(cbind(LB,UB),2))
## The 95% confidence interval for population mean is 15.81 18.59

Based on the 95% confidence interval of (15.81, 18.59), we are 95% confident that the true difference in mean score overestimation between men and women is between 15.81 and 18.59. Because all of the values on the interval are positive, we can conclude that there is evidence to support an association between male and female overestimation scores, which corroborates the conclusion from the p-value.

In summary, this study found that men, on average, overestimated their IQ by 7.7 points (+7.7) and women, on average, underestimated by 9.5 points (-9.5). The difference between these means (male minus female) was 17.2 points. The 95% confidence interval was (15.81, 18.59), which suggests that the true difference between men and women and their IQ estimation is likely positive, and not zero like the null hypothesis claims because zero is not included in the interval. The p-value of approximately zero and t-statistic of 24.31 both provide very strong evidence against the null hypothesis, which leads us to conclude that there is statistically significant evidence to suggest an association between gender and how much one overestimates their IQ, which agrees with the conclusions reached by previous studies.

In the future, if this study were to be conducted again, a few suggestions to improve the quality are recommended. Due to the relatively high subject withdrawal rate of 15% due to lack of timeliness in data collection, it would be prudent to record data as it is collected instead of waiting to write it down. Additionally, taking a simple random sample and allowing any town to be randomly selected may improve the level of representation for the study.

For future research, it may be interesting to try and conduct experiments based on the ideas presented in this study. For example, study the relationship between how much someone is socialized in weeks leading up to a test and how well they do on said test. Another option, in keeping with the gender association, is investigating whether gender affects human perception of other factors, such as math skill or workplace performance (that is, do men or women tend to think they’re better at one thing than another?). In conclusion, there are many relationships in the gender binary that may be interesting to look into further outside the scope of this singular study.

Agata Szymanowicz, Adrian Furnham, Gender and gender role differences in self- and other-estimates of multiple intelligences, The Journal of Social Psychology, Volume 153, 2013, Page 899423, (https://doi.org/10.1080/00224545.2012.754397)

David Reilly, David L Neumann, Glenda Andrews, Gender differences in self-estimated intelligence: Exploring the male hubris, female humility problem, Frontiers in Psychology, Volume 13, 2022, Pages 812-83, (https://doi.org/10.3389/fpsyg.2022.812483)