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 differences between paired observations: d i = x 1 i − x 2 i .
Compute the mean and standard deviation of these differences: d ˉ = 2 and s d ≈ 11.24 .
Find the critical t-value for a 99% confidence level with 6 degrees of freedom: t 0.005 , 6 = 3.707 .
Construct the 99% confidence interval: ( d ˉ − E , d ˉ + E ) = ( − 13.8 , 17.8 ) .
Explanation
Problem Analysis We are given two dependent random samples and asked to find the 99% confidence interval for the true difference between the population means. We assume that both populations are normally distributed.
Calculate Differences First, we need to calculate the differences between each pair of observations. The differences are calculated as follows:
d i = x 1 i − x 2 i for i = 1 , 2 , ... , 7
Using the given data, the differences are:
41 − 43 = − 2 38 − 32 = 6 30 − 45 = − 15 50 − 38 = 12 48 − 39 = 9 28 − 38 = − 10 36 − 22 = 14
So, the differences are: -2, 6, -15, 12, 9, -10, 14
Calculate Mean of Differences Next, we calculate the sample mean of the differences: d ˉ = 7 ∑ i = 1 7 d i = 7 − 2 + 6 − 15 + 12 + 9 − 10 + 14 = 7 14 = 2
So, the mean of the differences is 2.
Calculate Standard Deviation of Differences Now, we calculate the sample standard deviation of the differences: s d = 7 − 1 ∑ i = 1 7 ( d i − d ˉ ) 2
First, we calculate the squared differences from the mean:
( − 2 − 2 ) 2 = ( − 4 ) 2 = 16 ( 6 − 2 ) 2 = ( 4 ) 2 = 16 ( − 15 − 2 ) 2 = ( − 17 ) 2 = 289 ( 12 − 2 ) 2 = ( 10 ) 2 = 100 ( 9 − 2 ) 2 = ( 7 ) 2 = 49 ( − 10 − 2 ) 2 = ( − 12 ) 2 = 144 ( 14 − 2 ) 2 = ( 12 ) 2 = 144
Sum of squared differences = 16 + 16 + 289 + 100 + 49 + 144 + 144 = 758
Then, we divide by n − 1 = 7 − 1 = 6 :
6 758 = 126.333...
Finally, we take the square root:
s d = 126.333... ≈ 11.24
So, the standard deviation of the differences is approximately 11.24.
Find Critical t-value We are given that we need to find the 99% confidence interval. This means that α = 1 − 0.99 = 0.01 , so α /2 = 0.005 . The degrees of freedom are n − 1 = 7 − 1 = 6 . From the t-table, the critical t-value for a 99% confidence level with 6 degrees of freedom is t 0.005 , 6 = 3.707 .
Calculate Margin of Error Now, we calculate the margin of error:
E = t α /2 , n − 1 ⋅ n s d = 3.707 ⋅ 7 11.24 ≈ 3.707 ⋅ 2.646 11.24 ≈ 3.707 ⋅ 4.248 ≈ 15.75
So, the margin of error is approximately 15.75.
Construct Confidence Interval Finally, we construct the 99% confidence interval for the true difference between the population means:
( d ˉ − E , d ˉ + E ) = ( 2 − 15.75 , 2 + 15.75 ) = ( − 13.75 , 17.75 )
Rounding to one decimal place, the 99% confidence interval is (-13.8, 17.8).
Final Answer The 99% confidence interval for the true difference between the population means is ( − 13.8 , 17.8 ) .
Examples
Consider a scenario where you want to compare the effectiveness of two different teaching methods on student test scores. You administer both methods to two groups of students, ensuring that the students are paired based on similar academic backgrounds to reduce variability. By calculating the 99% confidence interval for the difference in means, you can determine the range within which the true difference in the effectiveness of the two methods likely lies. This helps in making informed decisions about which teaching method to implement, with a high degree of confidence.
A current of 15.0 A flowing for 30 seconds generates a total charge of 450 C. This charge results in the flow of approximately 2.81 x 10^21 electrons through the device. Therefore, 2.81 quintillion electrons flow during this time period.
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