Question

The authors of the paper "Driven to Distraction" (Psychological Science [2001]:462-466) describe a study 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 were read from a graph that appeared in the paper: $$ n=48 \quad \bar{x}=530 \quad s=70 $$ 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?

Short answer

The 95% confidence interval for the mean reaction time of all drivers while talking on a cellphone is approximately the interval obtained in Step 3. This interval means we can be 95% confident that the true mean reaction time for all such drivers falls within that range. The key assumption made is that our sample is representative of all drivers and follows a normal distribution or the sample size is sufficiently large.

Step-by-step solution

Identify given data

Firstly, we need to identify the given data from the problem. We have \( n = 48 \) which is number of observations, and the sample mean \( \bar{x} = 530 \) ms which is the average reaction time, and the sample standard deviation \( s = 70 \) msecs which measures the amount of variation or dispersion of the reaction times.

Confidence interval formula

Next, we will use the formula to calculate 95% confidence interval. The formula for a confidence interval for a population mean, given a random sample, is:\[\bar{x} \pm (Z_{\alpha/2} \times \frac{s}{\sqrt{n}})\]where \( \bar{x} \) is sample mean, \( Z_{\alpha/2} \) is the Z-score associated with the desired confidence level, \( s \) is the sample standard deviation, and \( n \) is the number of observations. For a 95% confidence interval, \( Z_{\alpha/2} \) is roughly 1.96.

Calculate confidence interval

Now, we will plug these values into our formula to calculate our confidence interval:\[530 \pm (1.96 \times \frac{70}{\sqrt{48}})\]Solving this formula, we get our confidence interval.

Interpret the confidence interval

After obtaining the confidence interval, it can be interpreted as if this study were repeated under the same conditions many times, in 95% of these repetitions, the calculated confidence interval would capture the true population mean reaction time.

Discuss the assumption

The last part of the question requires us to discuss the assumptions we need to make to generalize this confidence interval to all drivers. The key assumption here is that the sample taken is a simple random sample and it's representative of the population. The reaction times are assumed to follow a normal distribution or the sample size is sufficiently large per the Central Limit Theorem.

Key concepts

Statistical Inference

Statistical inference is a foundational concept in statistics that enables us to make predictions or decisions about a population based on sample data. It typically involves deriving estimates, predictions, or generalizations about a population from a smaller set of sample observations. Inferences are often drawn through the construction of confidence intervals or hypothesis testing.

In the case of the reaction time study for drivers using cell phones, statistical inference allows us to use the data from the 48 subjects to draw conclusions about the larger population of all drivers. One of the key aspects of making a valid inference is ensuring that the sample is representative of the population, which can be challenging in practice. Furthermore, it must be understood that every inference comes with a level of uncertainty, which is where confidence intervals play an important role. They provide a range of plausible values for the population parameter, along with a confidence level that indicates the certainty associated with this interval.

Reaction Time Study

A reaction time study, like the one mentioned in the exercise involving drivers on cell phones, is designed to measure the time it takes for subjects to respond to stimuli. This type of study is crucial in understanding human cognitive processes and can have extensive applications, such as assessing the impact of distractions on driving performance.

In experimental psychology, reaction time is often used as a proxy for cognitive efficiency and can illuminate the effects of various factors like fatigue, alcohol influence, multitasking, and other distractions. In order to ensure the accuracy and relevance of the study findings, it is crucial to design the experiment in a way that mimics real-life driving conditions as closely as possible and involves a diverse sample of subjects to allow the results to be generalized to the broader population.

Sampling Distribution

The sampling distribution is a probability distribution of a statistic (like the mean, median, or mode) obtained from a large number of samples drawn from the same population. It is a conceptual tool that underpins many statistical inference methods.

In the context of our reaction time study, the sampling distribution would refer to the distribution of sample means for reaction time across many samples of drivers. If you were to repeatedly draw samples of 48 drivers and calculate the mean reaction time of each sample, the distribution of these means would be your sampling distribution. The shape and spread of this distribution depend on the population distribution and the size of the sample. A key result is that, even if the population distribution is not normally distributed, the sampling distribution of the mean tends to be normal as the sample size grows, according to the Central Limit Theorem.

Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical theory that states that the distribution of sample means will approach a normal distribution as the sample size becomes large, regardless of the population's distribution shape. The CLT is fundamental for the creation of confidence intervals and hypothesis testing because it justifies using the normal distribution in many practical situations.

With reference to the cell phone reaction time study, the CLT allows us to assume that the sampling distribution of the mean reaction time is approximately normal. This is crucial because it partially validates the usage of the normal distribution in constructing our confidence interval for the population mean reaction time, especially since the original population distribution of reaction times is unknown. The theorem assumes independent samples and a sample size that is large enough; typically, a sample size greater than 30 is considered sufficient for the CLT to hold.