To find the correlation between the values, we first need to calculate the mean for both sets of data.
Mean of gestation periods:
(8 + 2.1 + 1.3 + 1 + 11.5 + 5.3 + 3.8 + 24.3) / 8 = 6.95
Mean of life spans:
(30 + 12 + 6 + 3 + 25 + 12 + 10 + 40) / 8 = 15.5
Next, calculate the covariance and standard deviation for both sets of data. Then use these to calculate the correlation coefficient.
Covariance:
Cov(X,Y) = Σ[(X - mean of X)(Y - mean of Y)] / n-1
Cov(X,Y) = [(8-6.95)(30-15.5) + (2.1-6.95)(12-15.5) + ... + (24.3-6.95)(40-15.5)] / 7
Cov(X,Y) = 78.5357
Standard deviation of gestation periods:
(Σ[(X - mean of X)^2] / n-1)^0.5
St. dev. of gestation = 6.88
Standard deviation of life spans:
(Σ[(Y - mean of Y)^2] / n-1)^0.5
St. dev. of life spans = 12.3
Correlation coefficient:
Correlation = Cov(X,Y) / (St. dev. of gestation * St. dev. of life spans)
Correlation = 78.5357 / (6.88 * 12.3) = 0.9549
For the slope of the regression line:
Slope = Cov(X,Y) / St. dev. of gestation^2
Slope = 78.5357 / 6.88^2 = 1.97
For the y-intercept, we can use the formula:
y-intercept = mean of Y - slope * mean of X
y-intercept = 15.5 - 1.97 * 6.95 = 1.58
To find the predicted lifespan for an animal with a gestation period of 6 months, we use the regression line equation:
predicted lifespan = slope * 6 + y-intercept
predicted lifespan = 1.97 * 6 + 1.58 = 13.5 years.