Question

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 realise that they are impaired, thinking “I only had 1-2 drinks … I am OK to drive.” A sample of 5 drivers was chosen, and their reaction times (seconds) 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. Below is the data gathered from this study:

Driver 1 2 3 4 5

Before 6.15 2.86 4.55 3.94 4.19

After 6.85 4.78 5.57 4.01 5.72. (a)The two measurements are dependent. Explain? why?

(b)Provide an estimate of the mean difference in reaction times between the two measurements? (c)Calculate and interpret a 95% confidence interval for the mean difference in reaction times between the two measurements?

(d)Use a 5% level of significance and the following points to test the claim that reaction times before drinking two bears is lower than reaction times after drinking two bears.

(i) State the null and alternative hypotheses in symbolic form and in context.

(ii) Calculate the test statistic.

(iii) Identify the rejection region(s).

(iv) Clearly state your conclusions (in context). (e)What would the conclusion be if using a 1% level of significance? Justify your answer

Answer 1

a) The two measurements are dependent because for each driver we have the matched data of before and after drinking two beers .

The Calculation table is given below:

Driver	Before	After	d =B-A	d^2
1	6.15	6.85	-0.7	0.49
2	2.86	4.78	-1.92	3.6864
3	4.55	5.57	-1.02	1.0404
4	3.94	4.01	-0.07	0.0049
5	4.19	5.72	-1.53	2.3409
		Total	-5.24	7.5626

b) The an estimate of the mean difference in reaction times between the two measurements is

The variance is

c) Degrees of freedom = n-1 = 4

The critical value of t for 4 df at 5% significance level is 2.776

Therefore the 95% CI is

= (-1.941, -0.155)

Interpretation : We are 95% confident that after drinking two beers, the true mean reaction time increases, thus the drivers are impaired.

d)

i) Null Hypothesis , there is no difference in reaction time after drinking two beers

ALternative Hypothesis , the reaction time before driking two beers is less than the reaction time after drinking two beers

ii) Under H0, the test statistic is

iii) The critical value of t for 4 df at 5% significance level for left tailed test is -2.132

Reject Region : Reject H0 if t < -2.132

iv) Since test statisc falls in rejection region. Rejects H0

Hence we have wnough evidence that after drinking two beers the reaction time for drivers are increases.

e) Significance Level

The critical value of t for 4 df at 1% significance level is -3.747

Since test statistic does not falls in rejection region. Fail to Reject h0.

Hnece we donot have enough evidence that reaction times before drinking two bears is lower than reaction times after drinking two bears.