To calculate the variance of the weights of the starting lineup of the football team, we can use the following steps:
- Calculate the mean (average) weight.
- Calculate the squared differences from the mean.
- Compute the average of those squared differences.
Step 1: Calculate the mean
First, we need to sum the weights and then divide by the number of weights.
Weights: \[ 217, 235, 212, 253, 233, 171, 184, 301, 256, 297, 247, 226, 237, 241, 195, 204, 197, 214, 166, 309, 281, 303, 309, 299 \] Total number of weights (n): 24
Sum of weights: \[ 217 + 235 + 212 + 253 + 233 + 171 + 184 + 301 + 256 + 297 + 247 + 226 + 237 + 241 + 195 + 204 + 197 + 214 + 166 + 309 + 281 + 303 + 309 + 299 = 6101 \]
Mean: \[ \text{Mean} = \frac{6101}{24} \approx 254.2083 \]
Step 2: Calculate squared differences from the mean
Next, we need to compute the squared difference for each weight from the mean.
\[ \begin{align*} (217 - 254.2083)^2 & \approx 1381.6486 \ (235 - 254.2083)^2 & \approx 374.0203 \ (212 - 254.2083)^2 & \approx 1774.6482 \ (253 - 254.2083)^2 & \approx 1.4465 \ (233 - 254.2083)^2 & \approx 447.0835 \ (171 - 254.2083)^2 & \approx 6863.6882 \ (184 - 254.2083)^2 & \approx 4908.9150 \ (301 - 254.2083)^2 & \approx 2209.9162 \ (256 - 254.2083)^2 & \approx 3.2220 \ (297 - 254.2083)^2 & \approx 1824.5521 \ (247 - 254.2083)^2 & \approx 51.0725 \ (226 - 254.2083)^2 & \approx 784.5158 \ (237 - 254.2083)^2 & \approx 293.3581 \ (241 - 254.2083)^2 & \approx 171.5444 \ (195 - 254.2083)^2 & \approx 3483.9126 \ (204 - 254.2083)^2 & \approx 2529.3782 \ (197 - 254.2083)^2 & \approx 3267.8660 \ (214 - 254.2083)^2 & \approx 1600.9668 \ (166 - 254.2083)^2 & \approx 7660.9675 \ (309 - 254.2083)^2 & \approx 3026.4139 \ (281 - 254.2083)^2 & \approx 710.5575 \ (303 - 254.2083)^2 & \approx 2392.4406 \ (309 - 254.2083)^2 & \approx 3026.4139 \ (299 - 254.2083)^2 & \approx 2010.1222 \ \end{align*} \]
Now we add all the squared differences together: \[ 1381.6486 + 374.0203 + 1774.6482 + 1.4465 + 447.0835 + 6863.6882 + 4908.9150 + 2209.9162 + 3.2220 + 1824.5521 + 51.0725 + 784.5158 + 293.3581 + 171.5444 + 3483.9126 + 2529.3782 + 3267.8660 + 1600.9668 + 7660.9675 + 3026.4139 + 710.5575 + 2392.4406 + 3026.4139 + 2010.1222 = 53639.816 \]
Step 3: Compute the variance
Since this is a sample from a population (the starting lineup), we will use the sample variance formula, which divides by (n - 1):
\[ \text{Variance} = \frac{53639.816}{24 - 1} = \frac{53639.816}{23} \approx 2332.1607 \]
Thus, the variance of the weights is approximately 2332.16 pounds².
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