91Ó°ÊÓ

In an experiment designed to study the effects of illumination level on task performance ("Performance of Complex Tasks Under Different Levels of Illumination," J. Illumin. Engrg., 1976: \(235-242)\), subjects were required to insert a finetipped probe into the eyeholes of 10 needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task. \(\begin{array}{lccccc}\text { Subject } & 1 & 2 & 3 & 4 & 5 \\ \text { Black } & 25.85 & 28.84 & 32.05 & 25.74 & 20.89 \\ \text { White } & 18.23 & 20.84 & 22.96 & 19.68 & 19.50 \\ \text { Subject } & 6 & 7 & 8 & 9 \\ \text { Black } & 41.05 & 25.01 & 24.96 & 27.47 \\ \text { White } & 24.98 & 16.61 & 16.07 & 24.59\end{array}\) Does the data indicate that the higher level of illumination yields a decrease of more than \(5 \mathrm{~s}\) in true average task completion time? Test the appropriate hypotheses using the \(P\)-value approach.

Step by step solution

01

Define Hypotheses

First, identify the hypotheses for this problem. We are testing if the true average time with white light (higher illumination) is less by more than 5 seconds compared to the time with black light.Null Hypothesis (H_0): \[ \mu_\text{Black} - \mu_\text{White} \leq 5 \]Alternative Hypothesis (H_a): \[ \mu_\text{Black} - \mu_\text{White} > 5 \]Here, \( \mu_\text{Black} \) and \( \mu_\text{White} \) represent the true average completion times for black and white backgrounds respectively.

02

Calculate Sample Differences

Compute the differences between completion times for each subject. For Subject 1: 25.85 - 18.23 = 7.62 For Subject 2: 28.84 - 20.84 = 8.00 For Subject 3: 32.05 - 22.96 = 9.09 For Subject 4: 25.74 - 19.68 = 6.06 For Subject 5: 20.89 - 19.50 = 1.39 For Subject 6: 41.05 - 24.98 = 16.07 For Subject 7: 25.01 - 16.61 = 8.40 For Subject 8: 24.96 - 16.07 = 8.89 For Subject 9: 27.47 - 24.59 = 2.88

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Key Concepts

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

Experimental Design

In this exercise, the experimental design plays a crucial role in understanding the impact of different illumination levels on task performance. Good experimental design ensures that the results are valid and reliable. The study involved measuring how long subjects took to complete a task under two different lighting conditions: low and high illumination. In this case, the objective was to determine whether higher illumination levels could improve task completion time significantly compared to lower levels.
The design involved a controlled setting with subjects performing a task repeatedly under both lighting conditions. This approach helps in minimizing sources of error and ensures that the variation in task completion time is primarily due to changes in illumination levels, rather than other factors.

• Each subject performs the task under both low and high illumination to reduce variability caused by individual differences.
• Randomization can be used to decide the order of conditions for each subject to mitigate any order effects.
• Control variables such as background color are kept constant except for the variable being tested (illumination level).

By setting up the experiment this way, researchers can confidently assess the impact of varying light conditions on task efficiency.

Illumination Effects

The main focus of this experiment is to analyze the effects of varied illumination levels on task performance. Illumination effects refer to how different levels of light influence the time it takes for subjects to perform a specific task.
In this study, subjects performed a task under low light with a black background and high light with a white background. These variations helped the researchers assess whether changes in illumination directly affected the speed of task completion.

• Higher illumination improves visibility, possibly reducing the time needed to complete visual tasks by diminishing errors and allowing faster operation.
• Darker conditions may lead to slower task performance due to increased difficulty in distinguishing task elements.
• The study’s results aim to show whether more light correlates with a quicker completion time and if this improvement surpasses a threshold of a 5-second decrease for practical significance.

Understanding these effects helps in optimizing work environments and could inform the design of spaces where visual tasks are critical.

Statistical Analysis

Statistical analysis allows researchers to make sense of the collected data by applying consistent methodologies to identify significant differences between groups. In this case, it aims to determine if the higher illumination level reduces the task completion time by more than 5 seconds on average.
The difference in completion times for each subject across the different lighting conditions is calculated. These differences are then statistically analyzed to check for significant differences using hypothesis testing.

• The average difference, or mean of the differences, gives insight into overall trends and efficiency gains.
• The standard deviation of those differences indicates variability, showing how consistently the change in illumination affects task time.
• The statistical test used should consider the magnitude and variance of these differences to decide if the observed change is due to chance or is a true effect of illumination.

Proper statistical analysis helps in drawing robust conclusions that are less likely to be influenced by random chance.

P-value Approach

The P-value approach is a statistical method used to make decisions regarding hypotheses in research. It quantifies the probability of observing data as extreme as, or more extreme than, the data collected if the null hypothesis is true. In simpler terms, it helps determine the strength of the evidence against the null hypothesis.
In this exercise, the null hypothesis posits that there is no significant decrease in the task completion time of more than 5 seconds when using higher illumination. The alternative hypothesis suggests the opposite: that the average time is indeed less by more than 5 seconds under high illumination.

• A small P-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting the alternative hypothesis might be true.
• Conversely, a large P-value suggests the data does not provide strong evidence against the null hypothesis, indicating that any observed differences could be due to random variation.
• P-values help in deciding whether the change in illumination is practically significant in reducing task completion time, aiding decision-making for lighting design.

This approach offers a clear criterion for decision-making in hypothesis testing, essential for driving scientifically-backed conclusions.