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 means of x (number of flowers) and y (total cost).
Calculate the standard deviations of x and y.
Calculate the covariance of x and y.
Calculate the correlation coefficient r using the formula r = s x ∗ s y co v ( x , y ) . The correlation coefficient is approximately 0.28 .
Explanation
Understanding the Problem We are given a table with the number of flowers in four bouquets and the total cost of each bouquet. Our goal is to find the correlation coefficient for the data in the table. The data points are (8, 12), (12, 40), (6, 15), and (20, 20).
Defining Variables and Formula Let x be the number of flowers and y be the total cost. We need to calculate the correlation coefficient r using the formula: r = s x ∗ s y co v ( x , y ) where co v ( x , y ) is the covariance of x and y, s x is the standard deviation of x, and s y is the standard deviation of y.
Calculating the Means First, let's calculate the means of x and y: x ˉ = 4 8 + 12 + 6 + 20 = 4 46 = 11.5 y ˉ = 4 12 + 40 + 15 + 20 = 4 87 = 21.75
Calculating Standard Deviation of x Next, we calculate the standard deviations of x and y: s x = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 s y = n − 1 ∑ i = 1 n ( y i − y ˉ ) 2
s x = 4 − 1 ( 8 − 11.5 ) 2 + ( 12 − 11.5 ) 2 + ( 6 − 11.5 ) 2 + ( 20 − 11.5 ) 2 s x = 3 ( − 3.5 ) 2 + ( 0.5 ) 2 + ( − 5.5 ) 2 + ( 8.5 ) 2 s x = 3 12.25 + 0.25 + 30.25 + 72.25 = 3 115 ≈ 6.184658
Calculating Standard Deviation of y s y = 4 − 1 ( 12 − 21.75 ) 2 + ( 40 − 21.75 ) 2 + ( 15 − 21.75 ) 2 + ( 20 − 21.75 ) 2 s y = 3 ( − 9.75 ) 2 + ( 18.25 ) 2 + ( − 6.75 ) 2 + ( − 1.75 ) 2 s y = 3 95.0625 + 333.0625 + 45.5625 + 3.0625 = 3 476.75 ≈ 12.607736
Calculating Covariance Now, we calculate the covariance of x and y: co v ( x , y ) = n − 1 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) co v ( x , y ) = 4 − 1 ( 8 − 11.5 ) ( 12 − 21.75 ) + ( 12 − 11.5 ) ( 40 − 21.75 ) + ( 6 − 11.5 ) ( 15 − 21.75 ) + ( 20 − 11.5 ) ( 20 − 21.75 ) co v ( x , y ) = 3 ( − 3.5 ) ( − 9.75 ) + ( 0.5 ) ( 18.25 ) + ( − 5.5 ) ( − 6.75 ) + ( 8.5 ) ( − 1.75 ) co v ( x , y ) = 3 34.125 + 9.125 + 37.125 − 14.875 = 3 65.5 ≈ 21.833333
Calculating Correlation Coefficient Finally, we calculate the correlation coefficient r: r = s x ∗ s y co v ( x , y ) = 6.184658 ∗ 12.607736 21.833333 ≈ 77.97531 21.833333 ≈ 0.279735
Final Answer The correlation coefficient is approximately 0.28. Therefore, the correct answer is 0.28.
Examples
Understanding correlation coefficients can be incredibly useful in real life. For example, a marketing analyst might want to know if there's a correlation between the amount spent on advertising and the sales revenue. If the correlation coefficient is high (close to 1), it suggests a strong positive relationship, meaning more advertising leads to more sales. Conversely, a correlation coefficient close to -1 would suggest that more advertising leads to fewer sales, which might indicate ineffective advertising strategies. A correlation coefficient close to 0 would suggest little to no linear relationship between advertising spend and sales revenue, implying other factors are influencing sales.
An electric device with a current of 15.0 A for 30 seconds allows approximately 2.81 billion billion electrons to flow through it, calculated using the charge formula and the charge of a single electron.
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