Question

Hi there, I have put up the full sheet but it is question two that I need answered the most. Thank you for your time.

Question 1.

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

1. The two measurements are dependent. Explain why. [1 mark]

2. Provide an estimate of the mean difference in reaction times between the two measurements. [4 marks]

3. Calculate and interpret a 95% confidence interval for the mean difference in reaction times between the two measurements. [15 marks]

4. 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.

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

(b) Calculate the test statistic.

(c) Identify the rejection region(s).

(d) Clearly state your conclusions (in context). [4 marks each]

5. What would the conclusion be if using a 1% level of significance? Justify your answer. [4 marks]

Question 2

This is part 2, this is the part that I need answered. Thank you for your time.

It was believed from the experiment on the obstacle course, in Part I, that there is a relationship between a subject’s reaction time before drinking two beers and the subject’s age:

Driver 1 2 3 4 5

Age (years) 20 30 25 27 26

1. What type of study is being outlined here? Justify your answer. [2 marks]

2. Plot a graph representing the relationship between reaction times before drinking two beers and age. [5 marks]

3. From the graph in Q2, suggest a relationship that could exist between the two measurements. [2 marks]

4. Use a 1% level of significance and the following points to test the claim that there is a relationship between the reaction times before drinking two beers and age.

(a) State the null and alternative hypotheses in context. [3 marks]

(b) Calculate the test statistic. [8 marks]

(c) Identify the rejection region(s). [4 marks]

(d) Clearly state your conclusions (in context). [4 marks]

5. What percentage of variation in reaction times before drinking two beers is unexplained by the relationship between reaction times before drinking two beers and age? [2 marks]

6. Derive a model/equation that could be used to predict reaction times before drinking two beers for a person, if the age of the person is known. [8 marks]

7. Using the model derived in Q6, what would the predicted reaction time, in the obstacle course, before drinking two beers of a 22-year-old be? [2 marks

Answer 1

Q2 Part 1

Type of study being outlined: Regression Study [cause-effect analysis] ANSWER 1

Reason being that an attempt is being made to see reaction time is influenced by the age of the driver. ANSWER 2

Q2 Part 2

The age is represented along the horizontal axis (x-axis) and reaction time along vertical axis (y-axis).

                       .

6
5						.
							.
4								.
3											.
2
1

                      20                                    25                                      30                                 

DONE

Q2 Part 3

Most likely relationship that could exist between the two measurements is linear ANSWER 3

Q2 Part 4

Let x = age of the driver and y = reaction time (in seconds) in an obstacle course before drinking 2 beers.

We postulate the relationship as: y = β₀ + β₁x + ε,

Part (a)

Null and alternative hypotheses: H₀: β₁ = 0 Vs H₀: β₁ ≠ 0 ANSWER 4

Part (b)

Test statistic: t = β_1cap/SE(β_1cap) = - 44.1561 ANSWER 5 [Details of calculations are given at the bottom.]

Part (c)

The rejection region: |t| > tcrit (t_{4, 0.005}) = 4.604 [using Excel Function] ANSWER 6

Part (d)

Conclusions

Since |t| > tcrit, H₀ is rejected.

So, we conclude that there is sufficient evidence to suggest that there is a linear relationship between age of the driver

and reaction time (in seconds) in an obstacle course before drinking 2 beers. ANSWER 7

Q5.

Percentage of variation in reaction times before drinking two beers unexplained by the relationship between

reaction times before drinking two beers and age = 1 – r² = 1- 0.9984 = 0.0016 = 1.6% ANSWER 8

Q6.

Equation that could be used to predict reaction times before drinking two beers for a person, if the age of the person is

known is: ycap = 12.6986 – 0.3264x ANSWER 8

7. Using the model derived in Q6, the predicted reaction time, in the obstacle course, before drinking two

beers of a 22-year-old would be: 5.5171 sec ANSWER

Details of Calculations

Data

i	xi	yi
1	20	6.15
2	30	2.86
3	25	4.55
4	27	3.96
5	26	4.19

Excel Calculation Summary

n	5
Xbar	25.60
ybar	4.342
Sxx	53.2
Syy	5.67748
Sxy	-17.366
β1cap	-0.3264286
β0cap	12.6985714
s^2	0.00290714
sb^2	5.4646E-05
r	-0.9992316
r^2	0.99846386
tb	-44.158142

DONE