Introduction

Driving under the influence of alcohol can be extremely dangerous for various reasons. One of the most noticeable effects of alcohol on humans is that it can impair reaction time. Below, we have a sample of 20 individuals who’s reaction times were measured before drinking two beers (“Before”) and after drinking two beers (“After”). The difference between these two variables (“AfterMinusBefore”) was calculated and graphed in the histogram below. The histogram is slightly positively skewed with more individuals with positive differences in reaction time. To see if alcohol really slows reaction time in this sample, a paired student t-test can be used to determine statistical significance. Paired means that we are comparing means of one individual in one group to themselves in a second group.

SubjectID Before AfterMinusBefore After
1 6.25 0.60 6.85
2 2.96 1.82 4.78
3 4.95 0.62 5.57
4 3.94 0.07 4.01
5 4.85 1.06 5.91
6 4.81 0.53 5.34
7 6.60 -0.51 6.09
8 5.33 0.51 5.84
9 5.15 -0.96 4.19
10 4.88 0.87 5.75
11 5.75 0.50 6.25
12 5.26 1.97 7.23
13 3.16 1.39 4.55
14 6.65 -0.23 6.42
15 5.49 -0.24 5.25
16 4.05 1.54 5.59
17 4.42 -0.46 3.96
18 4.99 0.94 5.93
19 5.01 1.02 6.03
20 4.69 -0.97 3.72

Statistical Difference?

Two Tailed Paired T-Test

Before we test to see if alcohol impairs reaction speeds, we can test to see if there is a general difference in the means with a two tailed t-test. The null hypothesis states that there is no difference between individual’s reaction times before and after drinking two beers.

\[H_0:\,\mu_d=\mu_1-\mu_2=0\] The alternative hypothesis states that alcohol does make a difference in individual’s reaction times before and after drinking two beers.

\[H_a:\,\mu_d\neq0\] When we set \(\mu_d\neq0\), we are only looking to see if there is any difference in the means in either the positive or negative direction.

From this, a two tailed paired t-test can be use to determine if, on average, the two beers each individual drank significantly changed their reaction times. To determine significance, an alpha of 0.05 will be used.

The results of the t-test below shows that the two beers each individual drank did significantly changed their reaction times (\(p<0.05\)) and we reject the null hypothesis. There is enough evidence to suggest that the two beers consumed changed the reaction times of the individuals.

t = 2.603, df = 19, p = 0.0175, mean diff = 0.504, 95% CI = [0.1, 0.91]

In addition, the t-test estimates that on average, each individual had a slower reaction speed after drinking two beers (\(\mu_d=0.504\)). Using the confidence interval, we can also see that the average reaction speed is somewhere between 0.1 and 0.91. Given these results, we can see if this slower reaction speed is statistically significant.

Right Tailed Paired T-Test

Since we are finding that this sample may have a slower reaction speed after drinking two beers, our new alternative hypothesis would be: \[H_a:\mu_d>0\] \(\mu_d\) is greater than zero because a positive difference between After and Before two beers (After - Before) would mean that reaction speeds after two beers were slower than before two beers. Note that we still keep the original null hypothesis which assumes there is no difference between these groups. The results of the right tailed t-test with an alpha of 0.05 is as follows:

t = 2.603, df = 19, p = 0.00873, mean diff = 0.504, 95% CI = [0.17, ∞)

With these results, there is enough evidence to suggest that drinking two beers impaired this sample’s average reaction speeds (\(p<0.05\)) and we reject the null hypothesis. It is highly unlikely that this mean difference before and after drinking two beers is due to chance.

Work Cited

The data for this report was sourced from: https://web.archive.org/web/20220519220227/https:/bolt.mph.ufl.edu/6050-6052/unit-4b/module-13/paired-t-test/