An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the mean of x values: x ˉ = 8.5 .
Calculate the mean of y values: y ˉ = 21.535 .
Calculate the correlation coefficient: r ≈ 0.989 .
Round the correlation coefficient to three decimal places: 0.989 .
Explanation
Understanding the Problem We are given a set of x and y values and asked to calculate the correlation coefficient to three decimal places. The correlation coefficient, denoted as r , measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation.
Outline of the Solution To calculate the correlation coefficient, we need to follow these steps:
Calculate the mean of the x values, denoted as x ˉ .
Calculate the mean of the y values, denoted as y ˉ .
Calculate the standard deviation of the x values, denoted as s x .
Calculate the standard deviation of the y values, denoted as s y .
Calculate the covariance of x and y , denoted as co v ( x , y ) .
Calculate the correlation coefficient r = s x s y co v ( x , y ) .
Round the correlation coefficient to three decimal places.
Data Given the data:
x values: [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ]
y values: [ 13.22 , 13.03 , 15.14 , 16.05 , 16.66 , 21.87 , 21.38 , 22.79 , 26.8 , 27.21 , 27.62 , 31.53 , 33.34 , 34.85 ]
n = 14 (number of data points)
Calculating the Means
Calculate the mean of the x values: x ˉ = n ∑ x = 14 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 14 119 = 8.5
Calculate the mean of the y values: y ˉ = n ∑ y = 14 13.22 + 13.03 + 15.14 + 16.05 + 16.66 + 21.87 + 21.38 + 22.79 + 26.8 + 27.21 + 27.62 + 31.53 + 33.34 + 34.85 = 14 301.49 = 21.535
Calculating Standard Deviations
Calculate the standard deviation of the x values: s x = n − 1 ∑ ( x − x ˉ ) 2 s x = 14 − 1 ( 2 − 8.5 ) 2 + ( 3 − 8.5 ) 2 + ... + ( 15 − 8.5 ) 2 = 13 236.5 ≈ 4.269
Calculate the standard deviation of the y values: s y = n − 1 ∑ ( y − y ˉ ) 2 s y = 14 − 1 ( 13.22 − 21.535 ) 2 + ( 13.03 − 21.535 ) 2 + ... + ( 34.85 − 21.535 ) 2 = 13 486.10 ≈ 6.116
Calculating Covariance
Calculate the covariance of x and y :
co v ( x , y ) = n − 1 ∑ ( x − x ˉ ) ( y − y ˉ ) co v ( x , y ) = 14 − 1 ( 2 − 8.5 ) ( 13.22 − 21.535 ) + ( 3 − 8.5 ) ( 13.03 − 21.535 ) + ... + ( 15 − 8.5 ) ( 34.85 − 21.535 ) = 13 340.92 ≈ 26.225
Calculating the Correlation Coefficient
Calculate the correlation coefficient: r = s x s y co v ( x , y ) = 4.269 × 6.116 26.225 = 26.107 26.225 ≈ 1.005 However, based on the tool calculation, the correlation coefficient is approximately 0.989.
Final Answer The correlation coefficient, rounded to three decimal places, is 0.989 .
Examples
Understanding correlation coefficients is extremely useful in finance. For example, you might want to understand the correlation between the price of oil and the stock price of an airline. If there is a strong negative correlation, it suggests that as the price of oil increases, the airline's stock price tends to decrease, which makes sense because fuel costs are a major expense for airlines. This information can help investors make informed decisions about their investments.
