In baseball, is therelinearcorrelation between batting average and home run percentage? Let x represent the batting average of a professional baseball player. Let y represent the home run percentage (number of home runs per 100 times at bat). Suppose a random sample of baseball players gave the following information x=0.251 0.259 0.29 0.265 0.269 y=1.3 3.7 5.8 3.9 3.7 my answer is strong and positive is this correct
From my calculations:
r = 0.923
Strong positive correlation.
To determine if there is a linear correlation between batting average (x) and home run percentage (y), we can calculate the correlation coefficient (r) and examine its sign and strength.
To calculate the correlation coefficient, you can follow these steps:
1. Calculate the mean of x and y. Let's call them x̄ and ȳ.
- x̄ = (0.251 + 0.259 + 0.29 + 0.265 + 0.269) / 5
- x̄ = 0.2668
- ȳ = (1.3 + 3.7 + 5.8 + 3.9 + 3.7) / 5
- ȳ = 3.28
2. Calculate the deviations of each x value and each y value from their respective means. Let's call them dx and dy.
- dx = (0.251 - 0.2668, 0.259 - 0.2668, 0.29 - 0.2668, 0.265 - 0.2668, 0.269 - 0.2668)
- dy = (1.3 - 3.28, 3.7 - 3.28, 5.8 - 3.28, 3.9 - 3.28, 3.7 - 3.28)
- dx = (-0.0158, -0.0078, 0.0232, -0.0018, 0.0022)
- dy = (-1.98, 0.42, 2.52, 0.62, 0.42)
3. Calculate the product of each dx and dy pair. Let's call it dxdy.
- dxdy = (-0.0158 * -1.98, -0.0078 * 0.42, 0.0232 * 2.52, -0.0018 * 0.62, 0.0022 * 0.42)
- dxdy = (0.031236, -0.003276, 0.058464, -0.001116, 0.000924)
4. Calculate the sum of dxdy.
- ∑dxdy = 0.031236 - 0.003276 + 0.058464 - 0.001116 + 0.000924
- ∑dxdy = 0.086232
5. Calculate the square of each dx and dy value. Let's call them dx^2 and dy^2.
- dx^2 = (-0.0158)^2, (-0.0078)^2, (0.0232)^2, (-0.0018)^2, (0.0022)^2
- dy^2 = (-1.98)^2, (0.42)^2, (2.52)^2, (0.62)^2, (0.42)^2
- dx^2 = (0.00024964, 6.084e-05, 0.00053824, 3.24e-06, 4.84e-06)
- dy^2 = (3.9204, 0.1764, 6.3504, 0.3844, 0.1764)
6. Calculate the sum of dx^2 and dy^2.
- ∑dx^2 = 0.00024964 + 6.084e-05 + 0.00053824 + 3.24e-06 + 4.84e-06
- ∑dx^2 = 0.00090512
- ∑dy^2 = 3.9204 + 0.1764 + 6.3504 + 0.3844 + 0.1764
- ∑dy^2 = 10.0076
7. Calculate the square root of ∑dx^2 * ∑dy^2. Let's call it sqrt(∑dx^2 * ∑dy^2).
- sqrt(∑dx^2 * ∑dy^2) = sqrt(0.00090512 * 10.0076)
- sqrt(∑dx^2 * ∑dy^2) = sqrt(0.009059914752)
- sqrt(∑dx^2 * ∑dy^2) = 0.0953
8. Calculate the product of the square root of each ∑dx^2 and ∑dy^2 and dxdy.
- r = (∑dx^2 * ∑dy^2) * (∑dxdy)
- r = 0.0953 * 0.086232
- r = 0.00822289896
Based on the calculations, the correlation coefficient (r) for the given data is approximately 0.0082. As r is very close to zero, it indicates a weak correlation between batting average and home run percentage. The positive or negative sign does not affect the strength but indicates the direction of the correlation. Therefore, the correct interpretation would be "weak and positive correlation" rather than "strong and positive correlation."
To determine whether there is a linear correlation between batting average and home run percentage, we can calculate the correlation coefficient.
Here are the steps to calculate the correlation coefficient:
Step 1: Calculate the mean of each dataset (batting average and home run percentage).
Mean of batting average (x): (0.251 + 0.259 + 0.29 + 0.265 + 0.269) / 5 = 0.2666.
Mean of home run percentage (y): (1.3 + 3.7 + 5.8 + 3.9 + 3.7) / 5 = 3.28.
Step 2: Calculate the deviation of each data point from the mean.
Deviation of each x-value: (0.251 - 0.2666), (0.259 - 0.2666), (0.29 - 0.2666), (0.265 - 0.2666), (0.269 - 0.2666).
Deviation of each y-value: (1.3 - 3.28), (3.7 - 3.28), (5.8 - 3.28), (3.9 - 3.28), (3.7 - 3.28).
Step 3: Square each deviation.
Squared deviation of x-values: (0.000036), (0.000051), (0.000493), (0.000014), (0.000001).
Squared deviation of y-values: (3.24), (0.1764), (5.1844), (0.4096), (0.1681).
Step 4: Calculate the product of the x and y deviations.
Product of x and y deviations: -0.020464, 0.008978, -0.065062, -0.002556, -0.000188.
Step 5: Calculate the sum of the squared deviations and the product of deviations.
Sum of squared deviations of x: 1.000095 x 10^-8.
Sum of squared deviations of y: 8.3895.
Product of deviations: -0.080452.
Step 6: Calculate the correlation coefficient (r).
r = (sum of products of deviations) / sqrt((sum of squared deviations of x) * (sum of squared deviations of y))
r = -0.080452 / sqrt((1.000095 x 10^-8) * 8.3895)
r ≈ -0.080452 / sqrt(8.39012517 x 10^-8)
r ≈ -0.080452 / 0.000289515
r ≈ -0.277888.
From the calculated correlation coefficient (r ≈ -0.277888), we can conclude that there is a weak negative correlation between batting average and home run percentage. Therefore, your answer of "strong and positive" is not correct.