Introduction

Is there a difference in GPA between males and females? An ACT information brief found that male students received higher average ACT grades in mathematics and science despite having a lower average GPA (based on a report by Brian Jacob) in those same fields than female students. This difference is what prompted me to study the difference in average GPA by sex among the islanders. Based on these reports, I suspected there would be a small difference between the average GPAs of males and females.

Data Collection Methods

To perform our study, we randomly selected 48 islanders across all the islands who had completed the 12th grade. In order to record the GPA of each participant, we first converted their six classes’ letter grades (C- to A+) to a numerical value (1.7 to 4.3). These six values were then averaged and recorded as that participant’s average GPA. We then asked each participant to identify themselves as male or female, and recorded the response. All data collection could be completed with one visit to each participant. Of the 48 initially selected islanders, 3 did not consent to the study, and thus we could not collect data from them.

Descriptive Statistics

We will use a box plot to visualize the average GPAs of males and females next to each other.

bwplot(sex ~ gpa, horizontal = TRUE, data = gpas,box.ratio = 2)

Based on the box plot, we can see that there does appear to be a slight difference in means between the two populations. Specifically, females appear to have about 0.2 higher of a GPA than males among the sampled population. There does seem to be more variance between within male GPAs.

favstats(gpa ~ sex, data=gpas)

When looking at the numerical summary, we can see that the means are 0.199 apart, and that both standard deviations are similar.

Analysis of Results

To determine the significance of these results, we need to conduct statistical tests.

For our study, the population of interest is the residents of the islands and the parameter of interest is the difference in mean gpa between the sexes. Our hypotheses are as follows:

\(H_0:\mu_{m}- \mu_{f} =0\) There is no difference in GPA between the sexes

\(H_a: \mu_{m}- \mu_{f} \neq 0\) There is a difference in GPA between the sexes

We will use the standard \(p < 0.05\) to determine statistical significance, as that best minimizes the risk of errors. A type I error in this scenario would mean that there truly is a difference in means, but we do not have strong enough evidence to reject the null hypothesis. A type II error means that we do find evidence to reject the null hypothesis, but in reality there is no difference in means.

Our measurements can be considered a representative sample from adults across the islands, as all participants were chosen by simple random sample.

The appropriate standardized statistic to use is a T-statistic to compare two means. The validity conditions for a t-test are a normal distribution of means, similar variances within data, and independent data. We meet all of these conditions. Data is independent, variances are similar (0.572 vs 0.516) and we can assume that a distribution of sample means would be distributed normally as both categories have 20+ data points.

n.men<- 25
n.women<- 20

x.bar.men<- 3.146
x.bar.women<- 3.345
  
SD.men<- 0.583
SD.women<- 0.529

x.bar.diff<-x.bar.men-x.bar.women
SE.x.bar.diff<-sqrt(SD.men^2/n.men + SD.women^2/n.women)
t.stat<-x.bar.diff/SE.x.bar.diff
cat("standard error of the difference in sample means is",round(SE.x.bar.diff,2),"
standardized statistic t is",round(t.stat,2),"
p-value is ",round(pval(t.test(gpa~sex,data=gpas)),3))
## standard error of the difference in sample means is 0.17 
## standardized statistic t is -1.2 
## p-value is  0.238

A t-statistic of -1.2 means that the observed difference in averages is 1.2 standard deviations below zero A p-value of 0.238 means that, assuming the null hypothesis is true, we have a 23.8% chance to randomly sample an absolute difference in means of 0.198 or more. This p-value means that we do not have sufficient evidence to reject the null hypothesis, so we will continue to believe there is no significant difference in GPAs between males and females.

To further visualize these results, we can construct a confidence interval with 95% confidence, equal to our p<0.05 significance level.

confint(t.test(gpa ~ sex, data = gpas))

Our 95% confidence interval is <-0.136, 0.534>, which means that we are 95% confident that the true population difference in means between females and males is between -0.136 and 0.534. As 0 is included within this interval, it remains possible that the true population difference in means is 0.

Conclusion

Through this study, we could not obtain sufficient evidence to reject the null hypothesis; that there is no difference in mean GPA between the sexes. A theory-based T-test yielded a T-statistic of -1.2, which gives us a two-sided p-value of 0.238. This t-statistic and p-value are not strong evidence against the null hypothesis. These results can reasonably be generalized to the rest of all islanders who have completed the 12th grade, as we made use of random sampling. To improve the accuracy of this study, future iterations should make use of much larger sample sizes. A related study could be done on the difference in school drop-out rates between males and females.

Bibliography: references to literature mentioned in the introduction

Jacob, Brian. “Where the boys aren’t: Non-cognitive skills, returns to school and the gender gap in Higher Education.” Economics of Education Review, May 2002, https://doi.org/10.3386/w8964.

Buddin, Richard. “Gender Gaps in High School GPA and ACT Scores.” ACT Research & Policy, Mar. 2014, www.act.org

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