Questions: At Bats and Hits The data below show the number of three-base hits (triples) and the number of home runs hit during the season by a random sample of MLB teams. Is there a significant relationship between the data? Triples 41 37 21 25 33 18 Home runs 318 294 319 312 293 285

Solution Steps

The covariance between the number of triples (\(X\)) and home runs (\(Y\)) is calculated as: \[ \text{Cov}(X,Y) = 20.3 \] The standard deviations are: \[ \sigma_X = 9.2177, \quad \sigma_Y = 14.5979 \] The correlation coefficient (\(r\)) is given by: \[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = 0.1509 \]

The means of \(X\) and \(Y\) are: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 29.1667, \quad \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 303.5 \] The numerator for the slope (\(\beta\)) is calculated as: \[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 53214 - 6 \cdot 29.1667 \cdot 303.5 = 101.5 \] The denominator for the slope is: \[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 5529 - 6 \cdot 29.1667^2 = 424.8333 \] Thus, the slope (\(\beta\)) is: \[ \beta = \frac{101.5}{424.8333} = 0.2389 \] The intercept (\(\alpha\)) is calculated as: \[ \alpha = \bar{y} - \beta \bar{x} = 303.5 - 0.2389 \cdot 29.1667 = 296.5316 \] The equation of the line of best fit is: \[ y = 296.5316 + 0.2389x \]

Final Answer