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The paper "Driving Performance While Using a Mobile Phone: A Simulation Study of Greek Professional Drivers" (Transportation Research Part F [2016]: \(164-170)\) describes a study in which 50 Greek male taxi drivers drove in a driving simulator. In the simulator, they were asked to drive following a lead car. On one drive, they had no distractions, and the average distance between the driver's car and the lead car was recorded. In a second drive, the drivers talked on a mobile phone while driving. The authors of the paper used a paired-samples \(t\) test to determine if the mean following distance is greater when the driver has no distractions than when the driver is talking on a mobile phone. The mean of the 50 sample differences (no distraction - talking on mobile phone) was 0.47 meters and the standard deviation of the sample differences was 1.22 meters. The authors concluded that there was evidence to support the claim that the mean following distance for Greek taxi drivers is greater when there are no distractions than when the driver is talking on a mobile phone. Do you agree with this conclusion? Carry out a hypothesis test to support your answer. You may assume that this sample of 50 drivers is representative of Greek taxi drivers.

Step by step solution

01

Define the null and alternative hypotheses

Let \(\mu_1\) denote the mean following distance with no distractions and \(\mu_2\) denote the mean following distance while talking on a mobile phone. Null Hypothesis (H0): There is no difference in the mean following distances in both scenarios. \[ H_0 : \mu_1 - \mu_2 = 0 \] Alternative Hypothesis (H1): The mean following distance is greater with no distractions than with talking on a mobile phone. \[ H_1 : \mu_1 - \mu_2 > 0 \]

02

Compute the test statistic

The test statistic for a paired-samples t-test is given by: \[ t = \frac{\bar{d} - \mu_{d,H_0}}{(\frac{s}{\sqrt{n}})} \] Where \(\bar{d}\) is the sample mean difference, \(\mu_{d,H_0}\) is the hypothesized mean difference, \(s\) is the standard deviation of sample differences, and \(n\) is the number of paired observations. In this case, we have the following values: \(\bar{d} = 0.47\), \(\mu_{d,H_0} = 0\), \(s = 1.22\), \(n = 50\). Plugging these values into the formula, we get: \[ t = \frac{0.47 - 0}{(\frac{1.22}{\sqrt{50}})} \]

03

Calculate the test statistic value

\[ t = \frac{0.47}{(\frac{1.22}{\sqrt{50}})} \approx 3.26 \]

04

Determine the critical value and make a decision

Now we need to determine the critical value for our hypothesis test. As it is a one-tailed test (greater than in the alternative hypothesis), we need to find the critical value for a one-tailed t-test. Let's consider a significance level of \(\alpha = 0.05\). Using a t-distribution table or a calculator, find the critical value for \(49\) degrees of freedom (n-1) and a 0.05 significance level. The critical value for a one-tailed t-test with 49 degrees of freedom and \(\alpha = 0.05\) is approximately 1.68. Since our test statistic \(t \approx 3.26 > 1.68\), we can reject the null hypothesis in favor of the alternative hypothesis.

05

Conclusion

There is enough evidence at a 0.05 significance level to conclude that the mean following distance for Greek taxi drivers is greater when there are no distractions than when the driver is talking on a mobile phone. Thus, we agree with the authors' conclusion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, denoted as H₀, is a fundamental concept in statistical hypothesis testing. It asserts that there is no effect or no difference, and any observed deviation in the data is merely due to random chance.

In our exercise regarding the Greek taxi drivers, the null hypothesis states that the mean following distance is the same whether the drivers were distracted by a mobile phone or not. Symbolically, the null hypothesis is represented as \[ H_0 : \mu_1 - \mu_2 = 0 \] where \(\mu_1\) is the mean following distance with no distractions, and \(\mu_2\) is the mean following distance while talking on a mobile phone.

Establishing the baseline assumption of no effect allows researchers to use statistical methods to determine if the evidence strongly suggests otherwise, which would then warrant rejection of the null hypothesis in favor of an alternative hypothesis.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis, denoted as H₁ or H_a, posits that there is a difference, effect, or relationship. The alternative hypothesis is what researchers aim to support.

For our specific study on driving performance, the alternative hypothesis claims that driving performance—as measured by following distance—is affected by the use of a mobile phone. It proposes that the mean following distance is greater without the distraction of a mobile phone than with it. In mathematical terms, this is expressed as \[ H_1 : \mu_1 - \mu_2 > 0 \]

This directional hypothesis specifies not just a difference but the direction of that difference. The hypothesis posits that not being distracted leads to an increased mean following distance, which would be considered safer in real-world driving.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to make a decision about the null hypothesis. The type of test statistic depends on the test being performed; for the paired-samples t-test, the test statistic helps assess the difference between paired observations.

In our driving simulation study, the test statistic is computed using the following formula: \[ t = \frac{\bar{d} - \mu_{d,H_0}}{(\frac{s}{\sqrt{n}})} \] where \(\bar{d}\) is the mean of the differences for paired measurements (here, the following distance without distraction minus the following distance while talking on the phone), \(\mu_{d,H_0}\) hypothetically equals zero (as per the null hypothesis), \(s\) is the standard deviation of the differences, and \(n\) is the sample size.

The calculated t-value then helps us determine whether the observed mean difference is statistically significant compared to the null hypothesis—providing a bridge to making informed inferences about the population from our sample.

Significance Level

The significance level, represented by alpha (\(\alpha\)), is the threshold for determining statistical significance in hypothesis testing. It signifies the probability of rejecting the null hypothesis when it is actually true (Type I error). Common values for \(\alpha\) are 0.05, 0.01, or 0.10.

For the paired-samples t-test in the Greek taxi drivers study, a significance level of 0.05 is selected. This means there is a 5% risk of concluding that the mean following distance is different when using a mobile phone—even if, in reality, there is no such difference (false positive).

If the calculated test statistic exceeds the critical value determined by the significance level, the null hypothesis is rejected. As described in the solution, with our test statistic of approximately 3.26 and the critical value for a one-tailed t-test at 1.68 for \(\alpha = 0.05\), we reject the null hypothesis, thus supporting the alternative hypothesis that drivers maintain a greater following distance when not distracted by a mobile phone.