Description and Source of Data

This material was adapted from the UF Biostatistics open Open Learning Textbook. https://bolt.mph.ufl.edu/6050-6052/unit-4b/module-13/paired-t-test/

Project Background

Drunk driving is one of the main causes of car accidents. Interviews with drunk drivers who were involved in accidents and survived revealed that one of the main problems is that drivers do not realize that they are impaired, thinking “I only had 1-2 drinks … I am OK to drive.”

A random sample of 20 drivers was chosen, and their reaction times in an obstacle course were measured before and after drinking two beers. The purpose of this study was to check whether drivers are impaired after drinking two beers.

before <- read.csv("Before2Beers.csv")
after <- read.csv("After2Beers.csv")
view(before)
view(after)
before_after <- merge(before, after)
view(before_after)
for( i in 1:nrow(before_after)){
  before_after$Difference <- before_after$After - before_after$Before
}
view(before_after)

Dataset with Calculated Differences

  1. The data table below contains the reaction time of 20 drivers before having two beers, their reaction time after having two beers, and the difference between their reaction time after two beers and before two beers.
before_after
##    SubjectID Before After Difference
## 1          1   6.25  6.85       0.60
## 2          2   2.96  4.78       1.82
## 3          3   4.95  5.57       0.62
## 4          4   3.94  4.01       0.07
## 5          5   4.85  5.91       1.06
## 6          6   4.81  5.34       0.53
## 7          7   6.60  6.09      -0.51
## 8          8   5.33  5.84       0.51
## 9          9   5.15  4.19      -0.96
## 10        10   4.88  5.75       0.87
## 11        11   5.75  6.25       0.50
## 12        12   5.26  7.23       1.97
## 13        13   3.16  4.55       1.39
## 14        14   6.65  6.42      -0.23
## 15        15   5.49  5.25      -0.24
## 16        16   4.05  5.59       1.54
## 17        17   4.42  3.96      -0.46
## 18        18   4.99  5.93       0.94
## 19        19   5.01  6.03       1.02
## 20        20   4.69  3.72      -0.97

Distribution

  1. The distribution of the histogram below appears to be roughly symmetrical, with possible slight skewness to the right, but it is not enough to determine it is not normal. Therefore, since there is not enough evidence to show it is not normal, it is safe to say the distribution is normal.
ggplot(before_after,aes(x=Difference))+geom_histogram(color = "black", fill = "light grey", bins = 8)

Paired Sample \(t\)-Test

  1. The mean difference in reaction times is 0.5035.
mean(before_after$Difference)
## [1] 0.5035
  1. \[H_0: d= 0\] \[H_a: d > 0 \]
t.test(before_after$Difference, alternative="greater",mu=0,conf.level = 0.98)
## 
##  One Sample t-test
## 
## data:  before_after$Difference
## t = 2.6031, df = 19, p-value = 0.008734
## alternative hypothesis: true mean is greater than 0
## 98 percent confidence interval:
##  0.07706723        Inf
## sample estimates:
## mean of x 
##    0.5035

  1. Reject the null.

  2. At 0.02 level of significance,there is enough evidence to conclude that a driver’s reaction time after two beers increases compared to their reaction time before the two beers.

Confidence Interval for the Differences in Reaction Times.

(11)Zero is not in the interval, which is consistent with our conclusion from the previous question. If zero was in the interval, it would mean we would fail to reject the null, but since it is, we still can reject the null using the confidence interval.

t.test(before_after$Difference,conf.level = 0.98)
## 
##  One Sample t-test
## 
## data:  before_after$Difference
## t = 2.6031, df = 19, p-value = 0.01747
## alternative hypothesis: true mean is not equal to 0
## 98 percent confidence interval:
##  0.01231381 0.99468619
## sample estimates:
## mean of x 
##    0.5035

Translating the Impact of Alcoholic Impairement into Real Terms

  1. The range for additional distance traveled by an impaired driver going 60mph, according to the confidence interval, is about 1.056 feet to about 87.56 feet.