Introduction

My research question seeks to investigate the relationship between gender and IQ levels. The research question is: ‘Is there a difference in mean IQ levels between males and females?’ I did the research on Ironbard island. The population parameter of interest is the difference in mean IQ test scores of male and female islanders aged 15 and above on Ironbard Island. The literature I draw provides insights from two relevant articles, Irwing and Lynn’s “Intelligence: Is there a sex difference in IQ scores?” and Furnham and Robinson’s “Sex Difference in Estimated Intelligence and Estimated Emotional Intelligence and IQ Scores.”

Irwing and Lynn’s research suggests a significant sex difference in general cognitive ability emerging at the age of 15, favoring males. Furnham and Robinson’s study further supports this by indicating that women tend to estimate their IQ lower than men, with the actual IQ scores showing a trend of lower averages for females. Hence based on these findings, there was a suspicion before observing any data, that the actual parameter value might show a higher average IQ for males compared to females among islanders aged 15 and above.

Data Collection Methods

I conducted a research on IQ difference tests on the Ironbard Island population. I chose Ironbard Island because it is a residential island with regular households, unlike other islands that may have churches and schools. I believe this island is simple and representative.The observational units in my study are adult residents of the island Ironbard who are 15 or older. The IQ score was measured by asking the islander to do an IQ test. It takes 20 minutes for each of the islanders to finish the IQ test and report the IQ score.

I used a simple random sampling approach to select these individuals to ensure that my sample was representative of the island’s population. There are 6 towns in total on Ironbard Island. So I assign each town a number from 1 to 6. I use a random number generator to give me a number between 1 to 6, and based on the number I find the island that matches with the number. I assigned each house a number, for example, in Vardo town there are 554 houses in total. I use a random number generator to give me a number between 1 to 554 and then select one islander who is over 15 years old from that house. I only select one islander from each house. If there are two or more people in that house who match the criteria, I will use the random generator again to select one individual for the study.

A few things that researchers need to be aware of is that if the islander who got randomly selected was younger than 15, then I will have to redo this random sampling. Because I am investigating a population equal to or over 15 years old. Luckily this situation did not happen in my process of collecting data. My sample size is 40 in total with 20 females and 20 males. If the male reaches 20 first, and the next one is still a male islander, I will have to do it again because I am taking 20 males and 20 females. Same if it happens to females first. This happened during my process when there was an extra female sample, so I did it again and got a male sample.

During the random sampling, 5 islanders declined to participate in my research. Other than that there was nothing went wrong during the sampling and collecting of data.

Descriptive Statistics

Quantitative variable is the IQ. They seems very close and there is no clear sign to derive any relationship between the IQ and Gender.

library(ggplot2)

ggplot(Mini_Data, aes(x = Gender, y = IQ, fill = Gender)) +
  geom_boxplot() +
  labs(title = "IQ Scores by Gender",
       x = "Gender",
       y = "IQ Score")

Using ggplot, I can observe that the average IQ for males and females is very close. However, females exhibit a broader range of IQ scores compared to males. And from the table, I can see the difference of mean for female and male are close to zero. Which matches the graph. Also Female has larger sd, which means that have a larger skew then Male. That also matches the graph.

favstats(IQ ~ Gender, data = Mini_Data)

Analysis of Results

The population of interest are the adult residents on Ironbard Island aged 15 and above. The parameters are the average IQ scores for male and female Ironbard islanders in this age group. The null hypothesis is that there is no significant difference between Male and Female’s average IQ in the age group for Ironbard islanders. \(H_0:\mu_{male}=\mu_{female}\) The alternative hypothesis is that the average IQ of Male will be higher that average IQ of Female in this age group for Ironbard islanders. \(H_a:\mu_{male} > \mu_{female}\)

Type I error in this case would be concluding that there is a significant difference in average IQ scores between genders but in fact there is no such difference. And type II error would be failing to find a significant difference in average IQ scores between male and female. When males do have a higher average IQ than females.

This measurements can be considered representative due to the random sampling method. The use of a random number generator for town and house selection across the entire island and redoing the sampling in case of an individual below 15 or been declined enhances the study’s representativeness. The sampling strategy provide generalizability of the study results to the larger population of adult residents on Ironbard Island aged 15 and above.

x.bar.f<-108.1
x.bar.m<-108.05
s.f<-23.5
s.m<-16.88
n.f<-20
n.m<-20

SE.null<-sqrt(s.f^2/n.f + s.m^2/n.m)
#SE.diff.null

t=(x.bar.m-x.bar.f)/SE.null
t
## [1] -0.00772813
-stat(t.test(IQ ~ Gender, data = Mini_Data))
##            t 
## -0.007723217

The standardized statistic t = -0.00772813 that the observed sample difference in mean IQ scores between male and female is 0.0077 standard errors below the mean of the null distribution. This is negative because male’s mean IQ is lower than female’s mean IQ. This value is the difference between (mean IQ of male - mean IQ of female)/ standard error for null hypothesis. The standardized statistic t=-0.00772813 reflects how many standard errors the sample mean is away from the null hypothesis mean. Because it is close to zero, that means the observed sample mean is very similar to what would be expected under the null hypothesis.

t.test(IQ ~ Gender, data = Mini_Data)
## 
##  Welch Two Sample t-test
## 
## data:  IQ by Gender
## t = 0.0077232, df = 34.48, p-value = 0.9939
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
##  -13.09998  13.19998
## sample estimates:
## mean in group Female   mean in group Male 
##               108.10               108.05

Then extract the p value by code.

pval(t.test(IQ ~ Gender, data = Mini_Data))
##   p.value 
## 0.9938823

From here we can find the right-tailed p-value by recognizing that the total area under the t-distribution curve equals to 1. p-value of 0.9939 from R output represents the two-sided p-value, i.e. area to the left of t=-0.007723 and area to the right of t=0.007723. Using this information we can then find the appropriate right-sided p-value (that would correspond to the area starting at t=-0.007723 to the right) as 1-0.9939/2=0.503. That means the probability of observing t = -0.00723 or greater is equal to 0.503 assuming null hypothesis is true. Based on this p-value, there isn’t enough statistical significance to conclude that there is a difference in average IQ between males and female. The direction of the evidence is in the opposite direction from the expected. I am fail to reject the null hypothesis because inconclusive results that goes in the different direction from the conjecture.

In the context of the problem and based on the statistical analysis, I fail to find evidence to conclude that there is a statistically significant difference in average IQ scores between males and females on Ironbard Island. The p-value of 0.503 exceeds the commonly used significance level of 0.05, leading us to fail to reject the null hypothesis. Therefore, I do not have statistical evidence to support that the average IQ scores is differ between the two gender groups.And we don’t have the evidence to support the alternative hypothesis. In this case the evidence is in the different direction.

The quantitative variable should have at least 20 observations in each group and the sample distributions should not be strongly skewed to meet a theory based approach validity conditions. In this case the study validity conditions for theory based approach are satisfied because the sample size was exactly 20 people per group and also seen from the box plot the distribution was not strongly skewed.

confint(t.test(IQ ~ Gender, data = Mini_Data))

We are 95% confidence that the true difference in mean IQ scores between males and females on Ironbard Island falls within the range of -13.09998 to 13.19998. This interval includes zero, indicating that at the 95% confidence level, there is no statistically significant difference in average IQ scores between the two gender groups. Also indicates a failure to reject the null hypothesis.

Conclusion

In conclusion, I was trying to explore the relationship between gender and IQ levels among residents aged 15 and above on Ironbard Island. From the data collected through a simple random sampling method and analyzed using both p-values and confidence intervals, both of them failed to provide a statistically significant difference in average IQ scores between males and females. That means I can conclude that they failed to reject the null hypothesis.

The p-value was 0.503 > 0.5. The 95% confidence interval for the difference in mean IQ scores ranged from -13.09998 to 13.19998, which contains zero and indicates a lack of statistically significance. This suggests that on Ironbard Island, gender does not appear to be a significant factor influencing average IQ scores in the specified age group.

The study has generalizability to the population with similar age on the island. There are also few limitations on this study, such as the relatively small sample size. To improve the study, future researchers can consider expanding sample sizes by random sampling method. Also try to explore additional factors (i.e cultural factors) on the island. A similar question can be what age group for male and female has higher IQ test score. Since we already know the average IQ are the same. So the research can try to figure out that are their IQ overall the same, or some age group has lower and another age group has higher and drive the average to be the same compare to another gender group.

What I learned from this study was how to conduct a observational research with the islander stimulator. Made me get used to how the real world research looks like, such as time consuming and potential decline from the participants. I was able to know how to apply the theory based approach and calculation to a quantitative response variable and binary categorical explanatory variable study. And draw from the study result, I realize that gender difference is not strongly related to IQ level.

Bibliography: references to literature mentioned in the introduction

Irwing, P., & Lynn, R. (2006). Intelligence: Is there a sex difference in IQ scores? Nature (London), 442(7098), E1–E1. (https://doi.org/10.1038/nature04966)

Furnham, A., & Robinson, C. (2023). Sex Difference in Estimated Intelligence and Estimated Emotional Intelligence and IQ Scores. The Journal of Genetic Psychology, 184(2), 133–144. (https://doi.org/10.1080/00221325.2022.2140025)

Lucas Zheng (2023),R pubs (http://rpubs.com/LucasZheng/1122234)