91Ó°ÊÓ

The authors of the paper "Driven to Distraction" (Psychological Science [2001]: \(462-466\) ) describe an experiment to evaluate the effect of using a cell phone on reaction time. Subjects were asked to perform a simulated driving task while talking on a cell phone. While performing this task, occasional red and green lights flashed on the computer screen. If a green light flashed, subjects were to continue driving, but if a red light flashed, subjects were to brake as quickly as possible and the reaction time (in msec) was recorded. The following summary statistics are based on a graph that appeared in the paper: $$n=48 \quad \bar{x}=530 \quad s=70$$ a. Construct and interpret a \(95 \%\) confidence interval for \(\mu\), the mean time to react to a red light while talking on a cell phone. What assumption must be made in order to generalize this confidence interval to the population of all drivers? b. Suppose that the researchers wanted to estimate the mean reaction time to within \(5 \mathrm{msec}\) with \(95 \%\) confidence. Using the sample standard deviation from the study described as a preliminary estimate of the standard deviation of reaction times, compute the required sample size.

Step by step solution

01

Calculate the 95% confidence interval for the mean

The 95% confidence interval for the mean is given by \(\bar{x} \pm Z_{\alpha/2} \times \frac{s}{\sqrt{n}}\). Here, \(\bar{x}\) is the sample mean given as 530 msec, \(s\) is the sample standard deviation given as 70 msec, and \(n\) is the sample size, which is 48. Since it's a 95% confidence, the Z-score (\(Z_{\alpha/2}\)) is approximately 1.96 (found from a standard Z-table). Plugging in these values, we get \(530 \pm 1.96 \times \frac{70}{\sqrt{48}}\).

02

Interpret the 95% confidence interval

The interval calculated in Step 1 gives a range of values in which you can be 95% confident that the true population mean falls in. However, to generalize this confidence interval to the population of all drivers, an important assumption that must be met is that the sample is representative of the population. If this is not the case, our confidence interval may not be applicable to the entire population.

03

Calculate the required sample size

The given researchers want to estimate the mean reaction time to within 5 msec with 95% confidence. The formula for the required sample size is \( n = \left(\frac{Z_{\alpha/2} \times s}{E}\right)^2 \). Here, \(E\) is the margin of error and is given as 5 msec, and the values of \(Z_{\alpha/2}\) and \(s\) are the same as before. Plugging these values, we get \( n = \left(\frac{1.96 \times 70}{5}\right)^2 \).

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Key Concepts

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

Reaction Time

Reaction time is a measure of how quickly someone can respond to a stimulus. In the context of the study "Driven to Distraction," reaction time was recorded during a simulated driving task where subjects needed to respond to red lights by braking. Measuring reaction time is crucial for understanding how different factors, such as distractions from using a cell phone, can impair a driver's ability to respond promptly. Factors affecting reaction time can include:

• Attention: Engaged attention generally leads to faster reaction times.
• Distraction: Other activities, like talking, can slow down reaction time.
• Fatigue: Tiredness often increases the time it takes to react.

Understanding reaction time is essential not only for safety in driving but also in various fields such as psychology and ergonomics. It helps uncover how multitasking or environmental factors might influence one's ability to respond to changes.

Sample Size Calculation

Sample size calculation is an important step in designing an experiment. It determines how many subjects need to be included to ensure the findings are statistically reliable. For this study, the researchers sought to ensure the mean reaction time could be estimated within a 5 msec margin of error. This necessitated calculating the appropriate sample size to achieve 95% confidence.To compute the necessary sample size, the formula used is:\[ n = \left(\frac{Z_{\alpha/2} \times s}{E}\right)^2 \]Where:

• \(Z_{\alpha/2}\) is the Z-score corresponding to the desired confidence level (1.96 for 95%).
• \(s\) is the sample standard deviation (70 msec in this study).
• \(E\) is the margin of error (5 msec).

Calculating using these values will give the required number of participants, ensuring the results are significant and reflective of the population they aim to represent.

Mean Reaction Time

Mean reaction time is the average time it takes for a subject to respond to a stimulus. In this study, the mean reaction time was found to be 530 milliseconds. Calculating the mean involves summing all the individual reaction times and dividing by the number of observations. The mean provides a central value that represents the typical reaction speed of the subjects under the conditions of the study. A few key points about mean reaction time include:

• It gives a summary statistic that can be quickly interpreted to understand the general trend.
• It serves as a point of comparison to determine how distractions, like talking on a phone, might impact cognitive functions.
• It's crucial for creating benchmarks in different conditions to evaluate the safety and effectiveness of drivers' responses.

Understanding mean reaction time is essential for assessing how an activity impacts performance and can help suggest improvements in driving safety strategies.

Standard Deviation

Standard deviation is a statistic that measures the spread of a dataset relative to its mean. In this study, a standard deviation of 70 msec was reported, reflecting how much the individual reaction times varied from the mean reaction time of 530 msec. A few important aspects of standard deviation include:

• Low standard deviation indicates that the data points tend to be close to the mean.
• High standard deviation suggests greater variability in reaction times.
• It helps understand the reliability of the calculated mean by providing context on data spread.

Standard deviation is vital in the interpretation of experimental data as it indicates consistency. In reaction time studies, analyzing standard deviation helps determine how consistently subjects perform the task, which is crucial for drawing conclusions about factors affecting reaction times.