Introduction

Our ability to react quickly to the events around us has been an important asset for humans since the very early days of our species.
From quickly registering a dangerous predator in the jungle to catching a piece of toast falling off your plate, we rely heavily on this natural instinct.
The average human conscious reaction time is considered to be around 0.2 for a visual stimulus, as stated by the SCI-FUN website.
This investigation attempts to test that and see whether a randomly sampled group of year 12 US students do not, on average, have a 0.2 reaction time.
The image below shows the process of reacting to visual stimulus

Introduction Cont.

The data used for this project is from the website Census at School - a United States classroom project that engages students in grades 4-12 in statistical problem solving.
Students complete a short online survey and the results are collated into a database.
One of the requirements of the survey is to complete a simple reaction time test.
Students must click stop when the square background colour changes and then record their time as a decimal on the survey.
The tests URL is: https://ww2.amstat.org/education/cas/2.cfm
Below are images of this test

Problem Statement

On average humans have a 0.2 reaction time to visual stimulus.
Assuming reaction times are normally distributed in the population, does this sample of year 12 US students provide evidence that the population mean is not actually 0.2 Seconds?
To find the answer the sample will be tested for normal distribution, its descriptive statistics will be gathered and the appropriate hypothesis test will be applied.
The results will then be properly interpreted at the end of this presentation.

Data

The data was taken from the Census at School website.
Students complete a short online survey and the results are collated into a database.
Users can then use the website’s random sampler form to access the results.
The URL is: https://ww2.amstat.org/censusatschool/RandomSampleForm.cfm
Here is an image of my selection:

Data Cont.

The data set used for this investigation was made with the Simple Random Sampling method
This is because the random sampler form takes a random selection of students from the database based on the selections made in the form.
For students to upload their survey they are required to enter in a Class ID and Class password, so there is some measure of data integrity.
The website URL is: https://ww2.amstat.org/censusatschool/index.cfm
The original data set contained 60 variables, ranging from foot length or dominant hand to what super power they would like and the number of text messages they received the day before.
For this investigation the data set will be reduced to only one variable: the reaction time.
Reaction_Time: numerical – the time from the colour change reaction test

Descriptive Statistics and Visualisation

The main variable is the students’ reaction time, this is a numerical variable that the students received after completing the colour change reaction test.
The data set contained a few issues, namely missing data and outliers.
Using the R function sum(is.na()) the total number of NA values were found and this amounted to 22.
To deal with this na.omit was used on the data set.

# reading in data set

students <- read.csv("1602471975663_DATA.csv")

str(students$Reaction_time)

##  chr [1:250] ".509" ".395" ".384" "0.5" ".386" ".346" "" ".69" ".39" ".761" ...

students$Reaction_time <- as.numeric(students$Reaction_time)

studentsTest <- students %>% select(Reaction_time)

# number of NAs

sum(is.na(studentsTest))

## [1] 22

# remove NAs

studentsTest <- na.omit(studentsTest)

sum(is.na(studentsTest))

## [1] 0

Decsriptive Statistics Cont.

A boxplot was used to investigate outliers.
While most reaction times sat under a second, there were five outliers which sat above this number.
They ranged from 1.089 to a staggering 398 seconds.
For this investigation we will be removing any outliers above the boxplots maximum, being any time above 0.9, on the assumption that those outliers are likely to be attributable to other factors (such as for example the student not paying full attention to the test).
Additionally, anything below 0.10 will be removed as around 0.15 seconds is considered to be the fastest possible conscious human reaction time, as stated by the Sci-Fun website. Some leeway is given because there is some uncertainty on this point.
Whether these were intentional errors or just data entry errors is unknown as students fill out the surveys themselves.
These outliers were removed from the data set using the filter() function

# reaction_time outliers and NAs
boxplot(studentsTest$Reaction_time)

which(abs(studentsTest$Reaction_time) >0.9)

## [1]  13  17  31  65 175 184 213

reaction_time_outliers <- studentsTest %>% filter(Reaction_time >1)

studentsTest2 <- studentsTest %>% filter(Reaction_time <0.9 & Reaction_time >0.10)

boxplot(studentsTest2$Reaction_time)

hist(studentsTest2$Reaction_time)

- While the distribution is skewed, we know the sampling distribution of a population mean with a sample size greater than 30, as is here n = 216, will be approximately normally distributed with a mean equal to the population mean.

Decsriptive Statistics Cont.

Here are the descriptive statistics for the reaction_time variable

studentsTest2 %>% summarise(Min = min(Reaction_time,na.rm = TRUE),
                                           Q1 = quantile(Reaction_time,probs = .25,na.rm = TRUE),
                                           Median = median(Reaction_time, na.rm = TRUE),
                                           Q3 = quantile(Reaction_time,probs = .75,na.rm = TRUE),
                                           Max = max(Reaction_time,na.rm = TRUE),
                                           Mean = mean(Reaction_time, na.rm = TRUE),
                                           SD = sd(Reaction_time, na.rm = TRUE),
                                           n = n(),
                                           Missing = sum(is.na(Reaction_time))) -> table
knitr::kable(table)

Min	Q1	Median	Q3	Max	Mean	SD	n	Missing
0.129	0.35475	0.4305	0.58525	0.891	0.4729306	0.1677707	216	0

As we can see the mean for Reaction_time is higher than 0.2

Hypothesis Testing

For the null hypothesis test we will assume that 0.2 seconds is the true average response time, and therefore our Null hypothesis will be 0.2 seconds and our alternate hypothesis will be that the mean is not equal to 0.2:

The Null hypothesis for this question is \[H_0: \mu = 0.2\sec\]

The Alternate hypothesis for this question is \[H_A: \mu \ne 0.2\sec\]

This investigation will be performing a one-sample t-test. This requires the population standard deviation to be unknown, which it is, and the data to be normally distributed in the population or the sample size is large (n>30).
As the sample size is well above 30, at a total of 216 after removing outliers and missing data, we can confirm the sample size is large.
The significance level will be α = 0.05.
Therefore we will be checking to see if the p-value is above 0.05. If it is not we can reject the Null hypothesis and consider the results to be statistically significant.
An upper tail critical value test will also be made.

Hypthesis Testing Cont.

The results of the t.test are as follows:

t.test(studentsTest2$Reaction_time, mu = 0.2)

## 
##  One Sample t-test
## 
## data:  studentsTest2$Reaction_time
## t = 23.909, df = 215, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0.2
## 95 percent confidence interval:
##  0.4504302 0.4954309
## sample estimates:
## mean of x 
## 0.4729306

As we can see the p-value for the hypothesis test was < 2.2e-16, well under the significance level of 0.5 and therefore we can reject the Null Hypothesis.
The results of the upper tail critical value test were as follows:

qt(p = .975, df = 216-1)

## [1] 1.971059

As we can see the upper tail critical value is 1.971 which is not captured by the 95 percent confidence interval of 0.450 to 0.495.
From the t-test and the upper tail critical value test we can reject the Null Hypothesis.

Discussion

The one-sample t-test was used to determine if the mean reaction times for year 12 US students were significantly different from the previously assumed reaction time mean of 0.2 seconds.
The 0.05 level of significance was used.
The sample’s mean reaction time was M = 0.473, SD = 0.168
The results of the one-sample t-test found the mean reaction time for year 12 US students to be statistically significantly different than the previously assumed reaction time mean.

Discussion Cont.

The strengths of this investigations were that the sample was made with the Simple Random Sampling method so there was no bias in gathering.
Also the sample size was large enough to conduct the one-sample t-test.
The weaknesses were that the surveys were not completed under strict conditions, and the outliers showed how some participants might have been affected by other factors.
For example, the students would have done the test on computers or mobile phones and the screens have lag which will have contributed to the reaction time.
For future investigation, the reaction times could be compared between males and females to see if sex is a factor in faster reaction times.

Average Reaction Time

Compared with School Students

RPubs link information

Introduction

Introduction Cont.

Problem Statement

Data

Data Cont.

Descriptive Statistics and Visualisation

Decsriptive Statistics Cont.

Decsriptive Statistics Cont.

Hypothesis Testing

Hypthesis Testing Cont.

Discussion

Discussion Cont.

References