Question

In past chapters of this study guide, you had the opportunity to learn about the Clock Test and do some calculations related to it. Here is the description of the test again if you need it. The Clock Test is a technique for studying human vigilance. In the Clock Test, a hand moves regularly at one step per second, but sometimes, at random intervals, it jumps two steps. The participant's task is to notice the two-step jumps and press a button.

The statistics that follow are the percentages of two-step jumps that were missed by 100 participants (N=100) during the first 15 minutes and the last 15 minutes of a two-hour watch.

Statistics on Time of Test:

First 15 min of watch: Sample Mean (X-bar) = 12 Sample Standard Deviation (s-hat) = 3.54

Last 15 min of watch: Sample Mean (X-bar) = 25 Sample Standard Deviation (s-hat) = 9.62

a. Calculate a 95% confidence interval about the mean for lapses during the first 15 minutes of a two-hour watch.

b. Calculate a 95% confidence interval about the mean for lapses during the last 15 minutes of a two-hour watch.

c. Write a short interpretation of your findings.

Answer 1

The 95% confidence intervals for lapses during the first and last 15 minutes of the vigilance test are (11.306, 12.694) and (23.115, 26.885) respectively. These intervals indicate that the participants' performance declined over the course of the test.

Step-by-step explanation:

The student is asking about constructing 95% confidence intervals for a set of data representing lapses during vigilance tests conducted at different times. To calculate the confidence intervals, we use the formula:

Confidence Interval = X-bar ± (Z*σ/√N)

Where X-bar is the sample mean, Z is the Z-score corresponding to the desired confidence level (for 95% confidence, Z is approximately 1.96), σ is the sample standard deviation, and N is the sample size.

1. First 15 minutes:
   Confidence Interval = 12 ± (1.96 * 3.54/√100)
   Confidence Interval = 12 ± (1.96 * 0.354)
   Confidence Interval = 12 ± 0.694
   Confidence Interval = (11.306, 12.694)
2. Last 15 minutes:
   Confidence Interval = 25 ± (1.96 * 9.62/√100)
   Confidence Interval = 25 ± (1.96 * 0.962)
   Confidence Interval = 25 ± 1.885
   Confidence Interval = (23.115, 26.885)

Interpretation:

The data suggest that participants were more vigilant during the first 15 minutes as they missed fewer two-step jumps compared to the last 15 minutes, shown by the lower mean and tighter confidence interval. As the watch progressed, the increase in the confidence interval range for the last 15 minutes reflects greater variability and indicates a decline in participants' performance, likely due to fatigue or loss of concentration.