An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate the mean times for both right-hand and left-hand threads: x ˉ r i g h t = 104.12 and x ˉ l e f t = 117.44 .
Find the mean difference between paired times: d ˉ = − 13.32 .
Compute the t-statistic: t = − 2.904 .
Determine the p-value and compare it to the significance level to conclude: There is a statistically significant difference.
Explanation
Understand the problem and provided data We are given data on the time it takes for 25 right-handed students to turn a knob with both a right-hand thread and a left-hand thread. Our goal is to determine if there's a statistically significant difference in the times required for each type of thread.
Calculate the means First, we calculate the mean time for the right-hand thread and the left-hand thread. The mean time for the right-hand thread is 104.12 seconds, and the mean time for the left-hand thread is 117.44 seconds.
x ˉ r i g h t = n ∑ x i = 104.12 x ˉ l e f t = n ∑ x i = 117.44
Calculate the mean difference Next, we calculate the mean of the differences between the paired observations (right thread time - left thread time). The mean difference is -13.32 seconds.
d ˉ = n ∑ ( x r i g h t , i − x l e f t , i ) = − 13.32
Calculate the standard deviation of the differences We then calculate the sample standard deviation of the differences, which is 22.936 seconds.
s d = n − 1 ∑ ( d i − d ˉ ) 2 = 22.936
Calculate the t-statistic Now, we calculate the t-statistic using the formula: t = s d / n d ˉ . The calculated t-statistic is -2.904.
t = s d / n d ˉ = 22.936/ 25 − 13.32 = − 2.904
Determine the p-value We determine the p-value associated with the calculated t-statistic and the degrees of freedom (n-1 = 24). For a two-tailed test, the p-value is 0.00779.
Compare the p-value and conclude Finally, we compare the p-value to a chosen significance level (e.g., 0.05). Since the p-value (0.00779) is less than the significance level (0.05), we reject the null hypothesis. This indicates that there is a statistically significant difference between the times for the right-hand thread and the left-hand thread.
State the final answer The analysis reveals a statistically significant difference in the times required to turn a knob with a right-hand thread versus a left-hand thread. Specifically, turning a knob with a right-hand thread is faster than turning a knob with a left-hand thread. Therefore, the final answer is that there is a statistically significant difference.
Examples
This type of analysis is crucial in ergonomics and human factors engineering. For example, when designing car dashboards or airplane cockpits, engineers need to ensure that controls are intuitive and easy to use. By conducting experiments and analyzing the data, they can determine which designs minimize reaction times and errors, ultimately improving safety and efficiency. This same approach can be applied to any situation where human interaction with controls is important, from medical devices to consumer electronics.
Approximately 2.81 x 10^{21} electrons flow through the device delivering 15.0 A of current for 30 seconds. This is calculated by determining the total charge and dividing by the charge of a single electron. The total charge is found to be 450.0 C.
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