An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Approximately 2.81 billion billion electrons flow through the device delivering 15.0 A of current for 30 seconds. This is calculated using the relationship between current, charge, and the charge of a single electron. The total charge that flows is 450 Coulombs, which corresponds to this number of electrons.
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Calculate the sample mean: n ∑ f x = 10 67 = 6.7 ≈ 7 .
Calculate the sample variance: n − 1 ∑ f x 2 − n ( ∑ f x ) 2 = 9 493 − 10 6 7 2 = 4.9 = 10 49 .
Calculate the standard deviation: 4.9 ≈ 2.214 .
The sample mean rounded to a whole number, the sample variance in reduced fraction, and the standard deviation rounded to three decimal places are 7 , 10 49 , and 2.214 respectively.
Explanation
Analyze the problem and given data We are given a table of class midpoints and frequencies, and our goal is to calculate the sample mean, sample variance, and standard deviation. We have the following data:
Sample size, n = ∑ f = 10
Sum of f ⋅ x , ∑ f x = 67
Sum of f ⋅ x 2 , ∑ f x 2 = 493
Calculate the sample mean First, let's calculate the sample mean. The formula for the sample mean is: Sample Mean = n ∑ f x Substituting the given values: Sample Mean = 10 67 = 6.7 Rounding the sample mean to a whole number, we get 7.
Calculate the sample variance Next, we calculate the sample variance. The formula for the sample variance is: Sample Variance = n − 1 ∑ f x 2 − n ( ∑ f x ) 2 Substituting the given values: Sample Variance = 10 − 1 493 − 10 ( 67 ) 2 = 9 493 − 10 4489 = 9 493 − 448.9 = 9 44.1 = 4.9 To express the sample variance as a reduced fraction: 4.9 = 10 49
Calculate the standard deviation Now, we calculate the standard deviation. The standard deviation is the square root of the variance: Standard Deviation = Sample Variance Standard Deviation = 4.9 ≈ 2.213594362117866 Rounding the standard deviation to three decimal places, we get 2.214.
State the final answer Therefore, the sample mean rounded to a whole number is 7, the exact value of the sample variance in reduced fraction is 10 49 , and the standard deviation rounded to three decimal places is 2.214.
Examples
Understanding sample mean, variance, and standard deviation is crucial in many real-world scenarios. For instance, in quality control, a manufacturer might measure the lengths of produced screws. The sample mean gives the average length, while the variance and standard deviation indicate the consistency of the production process. A high variance suggests that the screw lengths vary significantly, which might indicate a problem with the machinery. By monitoring these statistics, the manufacturer can ensure the quality and reliability of their products.
