Introduction

I looked at the Islander’s levels of conscientiousness and vocal frequency, begging the research question: Is there an association between levels of conscientiousness and vocal frequency? The population parameter I investigated was the long-run difference in mean vocal frequencies between those with high conscientiousness and low conscientiousness. I assumed that there would be a difference in the mean vocal frequency with differing levels of conscientiousness, but whether positively or negatively associated, I did not know. I suspected that the data would show a similar level of vocal frequency for the two groups.

Data Collection Methods

The observational units for my study were randomly selected participants from the island of Bonne Santé. A random number generator was used for the town number, and then a random house number based off of the total number of houses in the town was chosen. Following the choosing of the house, a person was randomly selected from the house depending on the number of residents. The response rate was about 70%, with many Islanders either declining to participate or being unable to give consent to participate. The levels of conscientiousness were always measured first, with each subject taking a 52-question personality test. A percentage was given for their level of conscientiousness, and those with higher than 50% were considered to be high conscientiousness and those with 50% or lower were considered to be low conscientiousness. The vocal frequency was measured through the participants saying “hello” and the frequency in Hertz was returned. One issue that arose through this collection was that there were many more people with high conscientiousness than low, and a total of 80 participants were needed to satisfy part of the validity conditions for the theory-based approach. There ended up being 60 participants with high conscientiousness and 20 participants with low conscientiousness.

Descriptive Statistics

Showing boxplots for the different levels of conscientiousness and summary statistics will help to show if there is a possible association between the mean vocal frequencies and the levels of conscientiousness. First, the data must be uploaded to be used for analysis.

library(readr)
Final <- read_csv("Copy of MATH 247 Mini Project 2 - Sheet1.csv")
head(Final, n = 2)

Boxplots:

bwplot(Conscientiousness ~ Frequency, horizontal = TRUE, data = Final)

Summary statistics:

favstats(Frequency ~ Conscientiousness, data = Final)

The level of conscientiousness is a binary categorical variable (higher or lower than 50% on the personality test). The vocal frequency is a quantitative variable. The boxplot shows very similar results for both high and low conscientiousness. The means, medians, first quartiles and third quartiles are very similar in terms of the frequencies in which they are at. However, the range of the high conscientiousness is much larger than that of the low conscientiousness, but there does not appear to be an association between the two observed variables. We can see through the summary statistics that the medians are identical, and the means only differ by 1.35 Hz. Additionally, the value of \(IQR_{high} = 72.25\) and \(IQR_{low} = 68.75\). These are very close as are the standard deviations between the two groups. It is evident that the difference in mean vocal frequency is very similar between those with high conscientiousness and those with low conscientiousness.

Analysis of Results

With the population being the residents of Bonne Santé and the population parameter being the long-run difference in mean vocal frequencies between those with high conscientiousness and low conscientiousness, the null and alternative hypotheses can be written as such:

Null: There is no difference between the mean vocal frequencies of those who have high conscientiousness and those who have low conscientiousness.

\[H_0: \mu_{high} - \mu_{low} = 0\]

Alternative: There is a difference between the mean vocal frequencies of those who have high conscientiousness and those who have low conscientiousness.

\[H_A: \mu_{high} - \mu_{low} \neq 0\]

A type I error would be rejecting the null that is actually true, which in this context would mean rejecting that there is no association when there is a true association. A type II error would be failing to reject the null when the null is false, which in this context means determining that there is no association when there is association between levels of conscientiousness and vocal frequency.

I believe that my measurements can reasonably be considered a representative sample from the population of interest. All participants were randomly sampled from the population and this random sampling can help to eliminate bias and approximate the true value if the entire population was measured.

The theory-based t-test requires that there are at least 20 observations in each category and that the data is not too skewed. We can see through the boxplots that the data is not too skewed and through the summary statistics that there are at least 20 observations.

stat(t.test(Frequency ~ Conscientiousness, data = Final))
##         t 
## 0.1376417

The resulting test statistic is \(t = 0.14\). This statistic is very small and therefore shows very weak evidence against the null.

The p-value corresponding to the alternative hypothesis will show further evidence about the decision to accept or reject the null. The p-value is the probability of observing a difference in mean vocal frequency as extreme or more extreme than than the observed statistic under a model that assumes that there is no difference between the mean vocal frequencies.

pval(t.test(Frequency ~ Conscientiousness, data = Final))
##   p.value 
## 0.8913123

This p-value is extremely large, and at a significance level of \(\alpha = 0.05\), shows weak evidence against the null. This means the null is plausible and should not be rejected. This means that there is not sufficient evidence to show an association between the mean vocal frequencies of those with high conscientiousness and low conscientiousness.

A confidence interval will further show information about the null and alternative hypotheses.

confint(t.test(Frequency ~ Conscientiousness, data = Final))

The 95% confidence interval gave a lower bound of -18.56 Hz and an upper bound of 21.26 Hz. This interval shows with 95% confidence the true range of the difference in frequency means of those with high conscientiousness and those with low conscientiousness. Zero is included in this confidence interval, which is further evidence that the null hypothesis cannot be rejected. This aligns with my result of the p-value. Both show strong evidence that the we fail to reject the null, thus there is not sufficient evidence to support the alternative hypothesis that there is a difference in mean vocal frequencies between high and low conscientiousness levels.

Conclusion

Based on evidence from both a theory-based p-value and a 95% confidence interval, there is no proven association between different levels of conscientiousness and vocal frequency. This is also shown visually through the boxplots, as the data is very similar between the two levels of conscientiousness. This is not what I expected, as I thought there would be some difference between the two. The data did not match my prediction. There was lots of consistency between the two levels and this showed throughout the analysis. It is not reasonable to say that the level of conscientiousness affects vocal frequency. I feel that this data could be generalized to the population of Bonne Santé, as having randomly sampled participants helps to eliminate errors and bias.

In terms of next time, I would sample differently and ideally have a larger sample size. This would help to eliminate Type II errors as well as make the data more valid. My data took a very long time to collect which is related to the smaller sample size. I only had 20 participants that had low conscientiousness which should be improved on for the future for more validated results. I learned some of the struggles of data collection, and can see how lengthy of a process real studies can be. If someone were to build on my results, they could look at how conscientiousness and gender relate, how other broad personality traits could affect vocal frequency, or how conscientiousness could impact other physical traits. Many different combinations of personality traits and physical traits can be put together to see if there is an association between them.